Enhanced Jaya Algorithm for design optimization of planar steel
frames
S. O. Degertekin1 I. B. Ugur2,
Department of Civil Engineering, Dicle University, 21100, Diyarbakır,Turkey
1
2
Department of Civil Engineering, Dicle University, 21100, Diyarbakır,Turkey
Abstract. An efficient meta-heuristic optimization method called Jaya Algorithm (JA) has gained wide acceptance
among optimization researchers in various engineering problems. The characteristic feature of JA is that does not
use algorithm-specific parameters and has a very simple formulation based on the concept of approaching the best
solution and moving away from the worst solution. This study presents an enhanced JA formulation for design
optimization of planar steel frames under strength, displacement and geometric-size constraints. The enhanced
Jaya Algorithm (EJA) accelerates the convergence to global optima and obtains better designs by randomizing a
specific number of best and worst solutions in each iteration in order to strengthen the exploitation behavior of the
method. The validity of EJA for planar steel frames is investigated by solving well-known five benchmark
problems up to 32 design variables. The results demonstrate the superiority of EJA over standard JA and other
state-of-the-art metaheuristic optimization methods in terms of optimized weight, number of structural analyses
and several statistical indices like standard deviation.
Keywords:
algorithm
design optimization; planar steel frames; meta-heuristic optimization methods; enhanced Jaya
1. Introduction
The emergence of meta-heuristic optimization methods that mimic natural phenomena has
increased considerably over the past three decades. A meta-heuristic could be defined as the
process of an iterative generation which sheds light on a heuristic by incorporating smartly
different concepts for exploration and exploitation of the search space and achieving strategies
in order to find near-optimum solutions [1]. Exploration and exploitation are the most
significant concepts of finding the best solution in all metaheuristic optimization methods.
Exploration provides generating diverse solutions in order to explore search space on a global
scale whereas exploitation focuses on the search in a local region by exploiting the information.
The balance between these two concepts allows to identify regions containing high-quality
solutions and move away from previously explored regions that are far from global optimum.
In the last two decades, the bio-inspired approaches (Genetic Algorithm (GA) [2], Particle
Swarm (PSO) [3], Ant Colony (ACO) [4], Honey Bee Mating (HBMO) [5], Enhanced Honey
Bee Mating (EHBMO) [6], Whale Optimization Algorithm (WOA) [7], Enhanced Whale
Optimization Algorithm (EWOA) [8] etc.) and physic-inspired approaches (Simulating
Annealing (SA) [9], Harmony Search (HS) [10], Big-bang big-crunch [11], Colliding Bodies
(CBO) [12] etc.) have been proposed for the optimization problems and extended by enhancing
their capabilities in optimization procedures such as the convergence, time consumption and
achieving the near-global optima.
Constructions must be designed as safely bearing the specified design loads constrained to
the structural provisions as well as the least amount of material is to be used. Meanwhile,
resource and time management are some of the most challenging problems in structural
engineering, however, structural designers can overcome these problems using meta-heuristics
to obtain the best design in terms of cost and safety. Since frame structures constitute the vast
majority of the skeletal systems in structural engineering, the optimum design of planar steel
frames is a common selected issue as a benchmark problem to investigate the efficiency of
meta-heuristics. Hence, various optimization methods have been proposed for the minimization
weight of steel frames under strength and stability circumstances specified in design codes.
Just to overview the literature published in the past two decades, Camp et. al. [13] used the
Ant Colony algorithm (ACO) that is to simulate the ant behavior to structural optimization of
steel frames.
Degertekin [14] utilized the Harmony Search (HS) based on the concept of searching for the
best harmony in musical improvisation. The efficiency of HS was tested in the optimization of
planar steel frames using three steel frames in comparison to the genetic algorithm and ant
colony optimization methods.
In the study proposed by Saka [15], structural optimization algorithms including GA, SA
and HS were reviewed and assessed comparing the optimization results of a steel frame design
example for each method.
Hasancebi et. al. [16] applied these techniques genetic algorithms, simulated annealing,
evolution strategies, particle swarm optimizer, tabu search, ant colony optimization and
harmony search to the real size rigidly connected steel frames including a planar example.
Dogan and Saka [17] developed an optimum design algorithm based on particle swarm
optimizer for planar steel frames. The superiority of the proposed algorithm was verified by
optimizing three steel frames in comparison to SA and GA.
An enhanced honey bee mating optimization method (EHBMA) for the optimum design of
side sway steel frames was proposed by Maheri and Nerimani [6] in order to overcome trapping
local optima and extend the search space of HBMO. The performance of the new method was
evaluated with four design examples.
Kaveh and Gaazan [8] proposed a new method called EWOA aimed to enhance the
convergence speed and solution accuracy of the standard method by modifying the formulation
of WOA. The efficiency of the new method was tested with four benchmark skeletal structures
involved steel frames and the results were compared to WOA and other optimization methods.
Carrero et. al. [18] implemented a search group algorithm (SGA) to three steel frame
examples in order to investigate the efficiency of the method in the structural design field. The
method has worked on the feasible domain only that leads to avoiding numerous structural
analyses. The results demonstrated that the proposed method achieved competitive
performance.
Farshchin et. al. [19] applied a school-based optimization (SBO) algorithm that is an
extended version of teaching-learning based optimization (TLBO) including multiple
classrooms and multiple teachers for the optimum design of planar steel frames
Most evolutionary and swarm-based intelligence algorithms require algorithm-specific
parameters for tuning the optimization process. However, inappropriate tuning of these
parameters affects the computational cost or convergence rate of the method negatively. In
order to overcome these issues, Rao [20] proposed a parameter-less evolutionary algorithm that
has a powerful search engine and can be easily implemented for any optimization problem. The
JA and its enhanced versions with various strategies have been utilized in large-scale real-life
urban traffic light scheduling problems [21], parameter estimating of battery models [22], cost
minimization of underground cable systems [23], structural damage detection [24] and medical
imaging problems. Besides, the JA was used also for the optimum design of truss structures
with both discrete and continuous variables [25,26]. The satisfactory performance of the JA in
optimum sizing of truss structures encouraged the authors to use JA in the structural
optimization of planar steel frames.
The main objective of this study is to minimize the weight of planar steel frames with
limitations of LRFD-AISC [27] and ASD-AISC [28] provisions using the JA and enhance the
standard JA in terms of faster convergence and stronger search capability. For this purpose, JA
is applied to sizing optimization of the well-known planar frame structures include up to 32
design variables utilized as the benchmark problems in the literature. In particular, the 244member planar frame is selected as an optimization problem since it has multiple search spaces
that vary according to the element type.
The remaining parts of the study are organized as follows. Section 2 recalls the discrete
sizing optimization of planar steel frames according to LRFD-AISC [27] and ASD-AISC [28].
Section 3 outlines the main steps for the implementation of the JA and EJA. Section 4 describes
the test problems and discusses optimization results. Section 5 provides a brief conclusion of
the study.
2. Sizing optimization of planar steel frames
To design an economical structure, the weight should be minimized within feasible
conditions. The weight of a steel structure depends on the material density, lengths and crosssectional areas of the elements. Cross-sectional areas are selected as design variables in the
optimization problem since material density and lengths are constant.
The objective of the optimization problem is to minimize the weight of structures under
stress, displacement and geometrical constraints by searching best cross-sectional areas in a
pre-defined section list. It can be stated as;
Find
𝐴 ∈ 𝑆 = {𝐴1 , 𝐴2 , . . . . . , 𝐴𝑛𝑐𝑠 }
ng
nm
k 1
i 1
To minimize W ( A)   Ak   i Li
k=1,2,….,ng
i=1,2,...,nm
(1)
j=1,2….nc
Subject to gj(A) ≤ 0
where A is the vector including the design variables (i.e. member groups), S is a set of
discrete cross-sectional areas, W(A) is the total weight of structure defined as a object function,
γi and Li are the material density and the length of member I , Ai is the cross-sectional area
of the member i., gj(A) denotes the design constraints including stress, displacement and
geometric constraints, n is number of constraints, ng is number of member group (i.e. design
variables), nm is the number of members, nd is the number of degrees of freedom, ncs is the
number of discrete cross-sectional areas in the section list.
In order to convert a constrained optimization problem to an unconstrained one, a
penalty approach is utilized to take into account the constraint violations. Accordingly, the
objective function is redefined as:
𝑊𝑝 (𝐴) = (1 + 𝜀1 ∙ 𝜓)𝜀2 × 𝑊(𝐴)
(2)
𝜓 = ∑𝑛𝑗=1 max⁡[0, 𝑔𝑗 (𝐴)⁡]
(3)
where n is the number of constraints for each design, 𝜓 represents the sum of the violated
constraints. 𝜀1 is the penalty constant set to 1, 𝜀2 is the exponent of the penalty function taken
as 2. The penalty parameters allow the objective function to approach in a feasible direction.
2.1 Constraints of frame structures
The optimum design of steel frame structures is subjected to displacement, stress and
geometrical constraints specified in the design codes. In this study, LRFD-AISC [27]
requirements are utilized for the first four examples whereas ASD-AISC [28] limitations are
selected as design constraints for the last example.
Strength constraints are described as following equations indicated in ASD-AISC for
different cases of fa/Fa:
𝑓
𝑓
𝑓
𝑓
𝑔𝑠,𝑗 (𝐴) = [𝐹𝑎 + 𝐹𝑏𝑥 + 𝐹𝑏𝑦 ] − 1 ≤ 0⁡⁡⁡⁡⁡⁡⁡𝑖𝑓⁡⁡ 𝐹𝑎 ≤ 0.15⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑗 = 1,2 … 𝑛𝑚
𝑎
𝑓
𝑔𝑠,𝑗 (𝐴) = [𝐹𝑎 +
𝑎
𝑏𝑥
𝑎
𝑏𝑦
𝐶𝑚𝑥 𝑓𝑏𝑥
𝑓
(1− 𝑎 )𝐹𝑏𝑥
𝐹′𝑒𝑥
+
𝐶𝑚𝑦 𝑓𝑏𝑦
𝑓
(1− 𝑎 )𝐹𝑏𝑦
𝐹′𝑒𝑦
𝑓
⁡] − 1⁡ ≤ 0⁡⁡⁡⁡⁡𝑖𝑓⁡⁡ 𝐹𝑎 > 0.15
𝑎
(4)
(5)
𝑓
𝑓
𝑓
𝑎
𝑔𝑠,𝑗 (𝐴) = [0.6𝐹
+ 𝐹𝑏𝑥 + 𝐹𝑏𝑦 ⁡] − 1 ≤ 0⁡⁡⁡⁡
𝑦
𝑏𝑥
(6)
𝑏𝑦
Fy is the yield stress of the material, fa is the required axial stress computed as P/A where A is
the cross-sectional area. Fa denotes the allowable axial stress depending on the elastic or
inelastic buckling behavior of the member. fbx and fby are required flexural stresses about the
major and minor axis. Fbx and Fby represent nominal flexural stress about the major and minor
axis. Fex and Fey stand for Euler stress on the -x and -y axis, respectively. Cmx and Cmy are
described as reduction factors that are used for the distribution of moments along the member
length depending on the sway behavior.
Strength constraints are described as following equations indicated in LRFD-AISC
against both bending and axial forces:
𝑃
8
𝑀𝑢𝑥
⁡⁡𝑔𝑠,𝑗 (𝐴) = 𝜙 𝑢𝑃 + 9 (𝜙
𝑏 𝑀𝑛𝑥
𝑐 𝑛
𝑃
𝑀𝑢𝑥
⁡⁡⁡𝑔𝑠,𝑗 (𝐴) = 2𝜙 𝑢𝑃 + (𝜙
𝑏 𝑀𝑛𝑥
𝑐 𝑛
𝑀𝑢𝑦
+𝜙
𝑏 𝑀𝑛𝑦
𝑀𝑢𝑦
+𝜙
𝑃
) − 1 ≤ 0⁡⁡⁡𝑖𝑓⁡ 𝜙 𝑢𝑃 ≥ 0.2⁡
𝑏 𝑀𝑛𝑦
𝑐 𝑛
𝑃
) − 1 ≤ 0⁡⁡⁡𝑖𝑓⁡ 𝜙 𝑢𝑃 < 0.2
𝑐 𝑛
(7)
(8)
Pu and Pn represent the required axial strength and the nominal axial strength for both
compression and tension; Mux and Mnx denote required flexural strength and nominal flexural
strength about the x-direction (major axis); Muy and Mny are the required flexural strength and
nominal flexural strength about the y-direction (minor axis). It should be noted that Mny=0 for
planar frame structures. 𝜙𝑐 is the axial resistance factor and taken as 0.90 for tension and 0.85
for compression; 𝜙𝑏 is the flexural resistance reduction factor and taken as 0.90.
Lateral displacement and inter-story drifts are considered as displacement constraints
and restricted to be less than the limit values described below:
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑔𝑑 (𝐴) =
∆𝑇
𝐻
− 𝑅 ≤ 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡
𝑑
𝑔𝑖𝑠,𝑛 (𝐴) = ℎ𝑛 − 𝑅𝐼 ≤ 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑛 = 1,2 … 𝑛𝑠⁡
𝑛
(9)
(10)
where ∆ 𝑇 represents the lateral displacement of the top story, H is the total height of the frame
structure, R denotes the maximum displacement limit taken as 1/300, dn is the inter-story drift
of the n-th floor, hn is the story height of the n-th floor. Ns is the number of stories. Rı denotes
the inter-story limit value specified as 1/300.
Apart from strength and displacement constraints, geometrical constraints are
considered so as to restrict the flange width of the beam section at each beam-column
connection to be less than or equal to the flange width of the column section. The equations of
geometrical constraints are shown below:
Figure 1. Beam-column connection
𝑔𝑔 (𝐴) =
𝑏𝑓𝑏
𝑏𝑓𝑐
𝑔𝑔 (𝐴) = (𝑑
− 1 ≤ 0⁡⁡
𝑏′𝑓𝑏
𝑐 −2𝑡𝑓 )
−1 ≤0
(11)
(12)
bf, 𝑏′𝑓𝑏 and bfc represent the flange width of the beam B1, the beam B2 and the column,
respectively, tf is the flange thickness of the column.
3. Enhanced Jaya Algorithm (EJA)
The JA quite recently developed optimization method is firstly proposed by Rao [20]. The
word “Jaya” originally means “victory” in Sanskrit. The algorithm is based on the concept that
the solution obtained for a given optimization problem should move toward the best solution
and must avoid the worst solution. The algorithm always tries to get closer to success (i.e.,
reaching the best design) and then tries to avoid failure (i.e., moving away from the worst
design) [20]. The most important feature of JA is that the JA has not any algorithm-specific
parameters whereas the other metaheuristic optimization algorithms have algorithm-specific
parameters. The JA only requires two standard control parameters which are the population size
(i.e., number of solutions in the population) and maximum iteration number.
The implementation of JA is very simple and has only one equation for modifying the designs.
Ak,l,it denotes the value of the k-th design variable for the l-th design during the it-th iteration, the JA
modifies the Ak,l,it as follows:



Aknew
,l ,it  Ak ,l ,it  r1, k ,it Ak ,best,it  Ak ,l ,it  r2, k ,it Ak , worst ,it  Ak ,l ,it

(13)
where Aknew
,l ,it is the new design variable for the Ak ,l ,it , r1,k ,it and r2 ,k ,it are the randomly generated real
numbers in the range [0,1] for the k-th design variable at the it-th iteration. Ak ,best ,it is the k-th design
variable of the best design at the it-th iteration and Ak ,worst,it is the k-th design variable of the worst


design at the it-th iteration. The term r1,k ,it Ak ,best,it  Ak ,l ,it indicates the tendency of the solution to


move closer to the best solution, and the term  r2,k ,it Ak ,worst,it  Ak ,l ,it indicates the tendency of the
solution to avoid the worst solution. It is worth pointing out that the random numbers r1 and r2 ensure
good exploration of the search space and the absolute value of the candidate solution (|Ak,l,it|)
considered in Eq (13) further enhances the exploration ability of the algorithm [20]. The flowchart
diagram of JA is shown in Fig.1.
There are many concepts are used in metaheuristic algorithms in order to strengthen the
searching process, converge the better global optima or accelerate the optimization. One of these
concepts is the elitism which is used during every generation in which elite (best) solutions replaced
with the worst solutions. In Enhanced Jaya Algorithm, unlike the elitism concept, worst solutions are
not duplicated by the elite ones. The specified number of best and worst solutions are identified.
Afterward, one of the identified solutions for each best and worst solutions is selected randomly then
used in the modification equation of JA. This process allows the algorithm to avoid the local trapping
and enhance the exploration capability.
As mentioned earlier, standard JA has no algorithm-specific parameter and only needs
common control parameters as population size (np) and maximum iteration number (itmax). The
optimization is terminated when the maximum iteration number is exceeded. However, each
optimization run could find the best solution for a different value of itmax. Instead, the following
formulation is implemented to terminate the search process when it is satisfied.
𝑆𝑇𝐷[𝑊𝑝 (𝐴1 ),𝑊𝑝 (𝐴2 ),…𝑊𝑝 (𝐴𝑛𝑠 )]
𝑛𝑠
∑
(
𝑖=1
1
≤ 𝜀𝑐𝑜𝑛
(14)
)
𝑊𝑝 (𝐴𝑖 )
In Eq. (14), 𝑊𝑝 (𝐴𝑖 ) represents the penalized objective function of i-th design in the population,
ns is the population size, 𝜀𝑐𝑜𝑛 is the coefficient of convergence tolerance taken as 10-4. As a
consequence of considering a dynamic itmax value, the needed common control parameters reduced to
only one which is population size.
4. Implementation of EJA for frame structures
In frame optimization, the EJA is initialized by randomly generated frame designs as the
population size np (number of frame designs) and the penalty functions for each design are
calculated by the results of structural analysis. The penalized functions are calculated at the rate
of constraint violation. After that, the frame design with a certain number of lower penalized
function values 𝑊𝑝 (𝐴𝑏𝑒𝑠𝑡
)
𝑖
and a certain number of higher penalized function values
𝑊𝑝 (⁡𝐴𝑤𝑜𝑟𝑠𝑡
) are assigned to the best design and the worst design vectors, respectively. The
𝑖
size of the best and worst vectors is set to 3. In order to modify the design variables by using
Eq.(13), the best and worst design is randomly selected from the best and worst vectors. The
new frame design is generated with modified design variables. The new penalized objective
function 𝑊𝑝,𝑛𝑒𝑤 (𝐴) is calculated. If the new penalized objective function value is smaller than
the previous one, ( 𝑊𝑝,𝑛𝑒𝑤 (𝐴) < 𝑊𝑝,𝑝𝑟𝑒 (𝐴) ), the new design is replaced with the previous one.
Otherwise, the previous design remains unchanged. This process is repeated for each frame
design in the population then an iteration is completed. The optimization is terminated when
the Eq. (14) is satisfied which depends on the standard derivation of penalized objective
function values in the population The Eq. (14) was defined as a termination criterion depends
on the standard deviation of penalized objective function values in the population and when the
criterion is satisfied, the optimization process is terminated. The best design without constraint
violation is reported as the optimum design.
The implementation of enhanced JA for optimum design of frame structures is summarized
below.
Step 1: Generate the initial population (i.e., frame designs) using random integers
between 1 and the number of discrete values for each design variable. Calculate
the penalized objective function values 𝑊𝑝 (𝐴) for all frame designs in the
population using Eq. (1-12). Set the iteration counter as it=0.
Step 2: Increase the iteration counter, it=it+1
Step 3: Determine the best and worst three designs of the population. 𝐴𝑖𝑏𝑒𝑠𝑡 , 𝐴𝑖𝑤𝑜𝑟𝑠𝑡 ⁡⁡𝑖 =
1,2,3. Select one of them as the best and worst design randomly.
Step 4: Modify discrete design variables by using Eq. (13) for each frame design in the
𝑛𝑒𝑤
population. Obtain the new designs with modified design variables⁡𝐴
𝑛𝑒𝑤
the penalized function values 𝑊𝑝 (𝐴
and calculate
) with modified design variables.
Step 5: If 𝑊𝑝 (𝐴𝑛𝑒𝑤
)<⁡𝑊𝑝 (𝐴𝑝𝑟𝑒
𝑖
𝑖 ) , replace the i-th new design with the previous one
otherwise; keep the previous design. Repeat this process for each frame designs
stored in the population.
Step 6: Terminate the optimization process if Eq. (14) is satisfied. Store the feasible
design corresponding to the lowest value of the objective function as the final
optimum design. Otherwise, go to Step 2
5. Design Examples
To demonstrate the performance the Enhanced Jaya Algorithm, five classical benchmark
frame examples as follows: Two-bay three-story steel frame, one-bay ten-story steel frame,
three-bay fifteen-story steel frame, a three-bay twenty four-story steel frame and 224-member
braced planar steel frame are optimized according to provisions of LRFD-AISC [27] and ASDAISC [28] specifications and the results are compared with standard Jaya Algorithm and other
metaheuristic methods in the literature.
The EJA and JA were executed ten times and the initial population of each run was
generated randomly. The population size was set to 20 for all examples.
The statistical performance and robustness of algorithms used in benchmark examples are
assessed and reported in related tables in terms of the best, mean, worst weighed designs, the
standard deviation of best designs obtained in each run and number of structural analyses. The
optimum design of each example reported in other studies was analyzed using the given
optimum sections in order to check constraint violation. If detected, the violation rates of both
strength and displacement constraints are reported in the tables.
EJA was coded in the MATLAB R2017a [29] programming language and automated to
interact with OPENSEES [30] structural analysis program for controlling design and
displacement constraints during the optimization process. The proposed program was executed
on a standard PC equipped with a single 2.2 GHz Intel® Pentium i7-4702MQ CPU.
5.1. Two-bay three story frame
The first design example to test the validity of EJA is two-bay three story frame as
shown in Fig. 2. The frame example has 15 members divided into 2 groups: beams and
columns. The column members are restricted to be selected from W10 sections and beams are
selected from all W shaped standard sections. The Young modulus of steel E is 29000 ksi and
yield stress is Fy is 36 ksi. For each column, the effective length factor is calculated according
to equations proposed by Dumonteil [31] for sway-permitted frames. The effective length
factor of the out-of-plane columns (Ky) is considered as 1.0. The unbraced length ratio of beams
is set to 0.167. Displacement constraints are not taken into account for this design.
Figure 2. Two-bay three story frame
This frame example was optimized by Camp et al. [13] using ACO, Degertekin [14]
using HS, Carraro et al. [18] using SGA, by Maheri and Nerimani [6] using HBMO and
Farshchin et al. [19] using SBO, JA and EJA from this study. The best design obtained by each
algorithm is reported as well as the statistical performance of these methods in terms of average
weight and standard deviation in Table 1.
The JA and EJA obtained the same design weighing 18792 lb as ACO, SGA and SBO
methods. While other methods required at least 420 structural analyses to converge the
optimum, JA and EJA required 155 and 72 structural analyses, respectively. It is evident that
the enhanced Jaya Algorithm converges the best design in much less structural analysis than
the others as shown in Fig 2. In ten independent runs, the mean weight and standard deviation
values of EJA are 18814 lb and 32.34 lb less than other methods except SBO. The results
reported by HS and HBMO methods are not feasible due to the constraint violations as
demonstrated in Table 1.
Table 1. Comparison of optimum designs for two bay three story frame
Optimal cross section area (in2)
Design
variables
ACO
[13]
HS
[14]
SGA
[18]
HBMO
[6]
SBO
[19]
JA
This study
EJA
This study
Column
W24×62
W21×62
W24×62
W10×77
W24×62
W24×62
W24×62
Beam
W10×60
W10×54
W10×60
W10×77
W10×60
W10×60
W10×60
18792
18292
18792
18000
18792
18792
18792
3000
1853
420
463
502
155
75
19163
18784
19011
N/A
18792
18851
18814
1693
411
385.09
N/A
0
36.68
32.34
Max IR
0.946
1.02
0.946
1.39
0.946
0.946
0.946
IRlimit
Max CV
(%)
1
1
1
1
1
1
1
None
2
None
39
None
None
None
Best
weight
(lb)
NSA
Mean
weight
(lb)
SD
Convergence curves are compared by plotting the structural weight versus number of structural
analyses in Fig 2. It is obvious that the EJA has the most powerful convergence ability among
all methods shown in Fig 2.
Figure 3. Comparison of convergence curves for the one-bay three story frame
5.2 One-bay ten story frame
A further design example commonly used in literature is one bay ten story plane as given
in Fig 4. with configuration, member grouping and load conditions. The frame has 30 members
which are gathered in 9 design groups and grouped in a way that the columns of two consecutive
stories and the beams of three consecutive stories except roof beams form a group distinctively.
Beam groups are selected from any 267 standard W-sections, column groups are restricted to
W12 and W14 sections. The material density, yield stress and the effective length factor of
members are considered as in the previous example. The columns are considered unbraced
along their length and the unbraced length for each beam member is specified as 1/5 of the span
length.
Figure 4. Two-bay three story frame
Two different optimization cases are considered in terms of displacement limitations. In
case 1, the inter-story drift for all stories is limited to story height/300; while in case 2, only the
displacement of top story is limited to total frame height/300. Both optimization cases are
considered in this study and the results are compared with other related methods in the literature.
Table 2. Comparison of optimum designs for one bay ten storey frame
Optimal cross section (area. in2)
Case 1
Sizing
variables
Case 2
GA
[34 ]
SBO
[19]
HBMO
[6]
EHBMO
[6]
1
2
3
4
5
6
7
8
W14×233
W14×176
W14×159
W14×99
W12×79
W33×118
W30×90
W27×84
W14×233
W14×176
W14×159
W14×99
W14×61
W33×118
W30×90
W27×84
W14×159
W14×159
W14×159
W14×159
W14×145
W21×73
W21×73
W24×76
W14×159
W14×159
W14×132
W14×109
W14×90
W24×94
W24×94
W18×76
JA
This
Study
W14x233
W14x176
W14x159
W14x99
W14x68
W33x118
W30x90
W24x84
EJA
This
Study
W14x233
W14x176
W14x159
W14x99
W12x65
W33x118
W30x90
W24x84
ACO
[13]
HS
[14]
SGA
[18]
SBO
[19]
W14×233
W14×176
W14×145
W14×99
W14×61
W30×108
W30×90
W27×84
JA
This
Study
W14x233
W14x176
W14x145
W14x99
W12x65
W30x108
W30x90
W27x84
EJA
This
Study
W14x233
W14x176
W14x145
W14x99
W12x65
W30x108
W30x90
W24x84
W14×233
W14×176
W14×145
W14×99
W12×65
W30×108
W30×90
W27×84
W14×211
W14×176
W14×145
W14×90
W14×61
W33×118
W30×99
W24×76
W14×233
W14×176
W14×132
W14×99
W14×68
W30×108
W30×90
W27×84
9
W24×55
W18×46
W24×76
W24×62
W21x44
W21x44
W21×44
W18×46
W21×50
W18×46
W21x44
W21x44
Best weight
(lb)
65136
64002
61014
57714
64298
64151
62610
61864
62262
62430
62610
62579
NSA
N/A
12691
1823
1340
18541
11857
8300
3690
7980
11677
7688
6933
Mean
weight (lb)
N/A
65880
N/A
N/A
66992.8
64469.5
63308
62923
65257
63244
62943
62684
Worst
weight (lb)
N/A
N/A
N/A
N/A
68785
65027.08
N/A
N/A
N/A
N/A
63518
63150
SD
Max ISD
N/A
0.0033
832
0.0033
N/A
0.00851
N/A
0.00596
1614
0.0033
294
0.0033
684
4.23
1.74
4.125
7027
4.224
706.84
4.217
340
4.23
252
4.32
ISDlimit
0.00333
0.0033
0.00333
0.00333
0.00333
0.00333
4.92
4.92
4.92
4.92
4.92
4.92
Max IR
0.967
1
1.91
1.438
0.968
0.998
0.986
1.0502
1.0148
0.999
0.986
0.998
IRlimit
1
1
1
1
1
1
1
1
1
1
1
1
Max CV (%)
None
None
155.1
78.9
None
None
None
5.02
1.48
None
None
None
The optimization performance of GA [34], SBO [19], HBMO [6], EHBMO [6] for case 1 and
ACO [13], HS [14], SGA, [18], and SBO [19] for case 2 is summarized in Table 2. Although
the best design of EJA is % 0.23 heavier than the SBO, EJA requires 834 less structural analysis
than SBO in case 1. Besides, EJA has the least standard deviation value that indicates the
robustness of the method. As detailed in the Table 2, best designs of HBMO and EHBMO are
extremely violated both strength and displacement constraints while EJA, JA, GA [34] and
SBO [19] strictly satisfy the limitations. In case 2, SBO [19] has the best design with the
weighing of 62430 lb that is % 0.24 lighter than EJA. It is clear that the best results of both
methods are rather close. In terms of required analysis number and statistical values, EJA has
the most satisfying performance with a standard deviation of 252 lb, a mean weight of 62684
lb and structural analysis number of 6933.
Figure 5.a. Comparison of convergence curves for the two-bay three story frame (Case 1)
Figure 5.b. Comparison of convergence curves for the two-bay three story frame (Case 2)
The convergence history of EJA and JA is indicated and compared with other methods
for both cases in Fig. 5. The convergence rate of the EJA is significantly higher than the others
as seen in the figure. In comparison to JA, the trend of achieving the near-optimum region is
faster by far.
5.3. Three Bay Fifteen Story Frame
Three bay fifteen-story planar frame was optimized firstly by Saka [15] using SA and
GA according to both LRFD [27] and ASD [28] provisions. The geometry and load conditions
of the frame are shown in Figure 6. The frame consists of 105 members divided into 12 design
groups. Grouping is considered as consecutive three-story inner and outer columns form a
distinct group, roof and intermediate story beams constitute a distinct group. The frame is
subjected to gravity loading as well as wind loading which computed according to The British
Code considering 45 m/s wind speed and 6 m frame spacing [15]. The modulus of elasticity
is 200 kN/mm2. In this example, both inter-story drift and lateral displacement of the top story
are considered as displacement constraints and restricted to be smaller than 1.17 cm and 17. 67
cm, respectively. In this example, the strength capacities of members are calculated with the
formulations specified in LRFD-AISC [27].
Figure 6. Three bay fifteen story frame
The optimum design results of EJA and other methods are reported in Table 4. The EJA
obtained the best design weighing of 33896 kg which is %9.27 lighter than the best design
obtained using PSO [17], %13.7 lighter than the SA and %17.22 lighter than GA [15]. In
addition, EJA has found better a design with less structural analysis with less standard deviation
than standard JA. The design of PSO [17] violated displacement constraint at a high rate as seen
in Table 3.
Table 3. Comparison of optimum designs for three bay fifteen storey frame
Sizing
variables
GA
[15]
SA
[15]
PSO
[17]
JA
This Study
EJA
This Study
1
W21×50
W21×50
W6×9
W8×18
W8×18
2
W24×55
W21×57
W21×44
W21×44
W21×44
3
W10×39
W10×33
W10×33
W12×35
W10×54
4
W14×53
W10×39
W10×33
W16×40
W16×36
5
W14×53
W12×53
W14×53
W21×55
W21×48
6
W14×68
W16×67
W21×111
W24×84
W21×68
7
W24×117
W24×104
W21×111
W30×90
W30×90
8
W14×43
W10×39
W14×61
W12×35
W8×24
9
W14×48
W14×48
W14×61
W16×40
W16×40
10
W14×68
W14×61
W24×76
W21×55
W24×62
11
W14×109
W14×99
W27×94
W30×90
W30×90
12
W16×100
W14×99
W27×102
W33×118
W30×108
40949
39262
37360
34103
33896
Best
(kg)
weight
NSA
25000
15500
7000
7870
7165*
Mean weight
(lb)
N/A
N/A
N/A
35381
34664
Worst weight
(lb)
N/A
N/A
N/A
37395
36752
SD
N/A
N/A
N/A
1366
1199
Max ISD
1.12
1.17
1.07
1.16
1.16
ISDlimit
1.17
1.17
1.17
1.17
1.17
Max IR
0.95
0.91
0.87
0.99
0.99
1
1
1
1
1
None
None
116
None
None
IRlimit
Max CV (%)
The design history graph of optimization using PSO [17], JA and EJA is plotted in Fig.
8. The convergence of EJA is rather satisfying in comparison to PSO [17] by far. JA has similar
convergence behavior with EJA as illustrated in Fig 7.
Figure 7. Comparison of convergence curves for the three-bay fifteen story frame
5.4 Three-Bay Twenty-Four Story Frame
The fourth, commonly used benchmark example is three-bay twenty-four story frame
consisting of 168 members that are collected in 20 groups shown in Fig. 8. with configuration
and loading conditions. The frame was originally designed by Davison and Adams [32] later
optimized by Camp et. al using PSO [13] , Degertekin using HS [14], Carraro et. al. using SGA
[18], Mahari and Nerimani using HBMO and EHBMO [6], Kaveh and Ghazaan using WOA
and EWOA [8] and Farshchin et. al. using SBO [19].
The material modulus of elasticity is 29782 ksi and the yield stress is taken as 33.4 ksi.
All members are considered unbraced along their lengths. For each column, the effective length
factor is calculated according to equations proposed by Dumonteil [31] for sway-permitted
frames. The effective length factor of the out-of-plane columns (Ky) is considered as 1.0. The
beam member groups could be selected from W-shaped sections in AISC standard list while
the column members are limited to W14 sections. The grouping scheme is demonstrated in Fig.
8.
Figure 8. Three-bay twenty-four story frame
Table 4 lists results of optimum designs including EJA and other optimization methods. The
EJA has the best design weighing of 201193 lb in comparison to all techniques reported in the
table. It should be noted that the EJA is overall the most efficient optimizer with the lightest
feasible design and the lowest standard deviation value. Much as SGA [18] and EHBMO [6]
found lighter designs weighing of 194508 and 188640 lb respectively, these designs violate
constraints at high rates as %34 and %1766.
Table 4. Comparison of optimum designs for three bay twenty four story frame
Optimal cross section (area. in2)
Sizing
variables
W30×90
JA
This
Study
W30×90
EJA
This
Study
W30×90
W10×30
W8×18
W16×26
W12×19
W21×62
W24×55
W21×48
W24×55
W24×55
W14×26
W6×8.5
W6×8.5
W6×8.5
W6×8.5
ACO
[13]
HS
[14]
SGA
[18]
HBMO
[6]
EHBMO
[6]
WOA
[8]
EWOA
[8]
SBO
[19]
1
W30×90
W30×90
W24×68
W10×22
W10×15
W30×90
W30×90
2
W8×18
W10×22
W21×55
W27×539 W36×256
W10×17
3
W24×55
W18×40
W24×62
4
W8×21
W12×16
W12×87
W8×21
W6×16
W33×221 W27×146
5
W14×145 W14×176 W14×159 W14×145 W14×145 W14×109 W14×159 W14×152 W14×120 W14×159
6
W14×132 W14×176 W14×145 W14×145 W14×120 W14×145
7
W14×132 W14×132 W14×120
W14×68
W14×26
W14×109 W14×120 W14×109
W14×90
W14×109
8
W14×132 W14×109
W14×99
W14×22
W14×26
W14×99
W14×74
W14×74
W14×90
W14×74
9
W14×68
W14×82
W14×68
W14×48
W14×53
W14×53
W14×74
W14×82
W14×74
W14×61
10
W14×53
W14×74
W14×48
W14×68
W14×99
W14×43
W14×43
W14×43
W14×34
W14×38
11
W14×43
W14×34
W14×48
W14×132 W14×159
W14×34
W14×30
W14×34
W14×30
W14×34
12
W14×43
W14×22
W14×34
W14×342
W14×22
W14×22
W12×19
W14×22
W14×22
13
W14×145 W14×145 W14×109 W14×159 W14×145 W14×120
W14×90
W14×109
14×109
W14×90
14
W14×145 W14×132
14×109
W14×99
15
16
W14×30
W14×99
W14×120 W14×120 W14×132
W14×82
W14×109
W14×26
W14×99
W14×120 W14×109
W14×120 W14×109
W14×99
W14×99
W14×74
W14×109
W14×90
W14×99
W14×109
W14×90
W14×90
W14×82
W14×109
W14×48
W14×26
W14×82
W14×99
W14×99
W14×90
W14×90
17
W14×90
W14×61
W14×90
W14×43
W14×26
W14×90
W14×68
W14×68
W14×74
W14×74
18
W14×61
W14×48
W14×74
W14×53
W14×26
W14×61
W14×61
W14×61
W14×74
W14×61
19
W14×30
W14×30
W14×43
W14×176 W14×370
W14×38
W14×43
W14×34
W14×43
W14×34
20
W14×26
W14×22
W14×43
W14×211 W14×109
W14×22
W14×22
W14×22
W14×22
W14×22
220465
214860
194508
214848
188640
206520
203490
202422
203069
201193*
15500
13924
8010
2074
1826
19640
18820
14572
13097
16306
229555
222620
213545
N/A
N/A
216475
208648
209560
207949.2
203612.9
N/A
N/A
N/A
N/A
N/A
243143
226019
N/A
216308.9
207178.9
4561
N/A
7027
N/A
N/A
N/A
N/A
7052
4204
1966
0.00307
0.0329
0.00447
0.0331
0.0622
0.00332
0.00332
0.0032
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.0033
0.00333
0.00333
Max IR
0.779
0.0774
0.949
4.69
10.75
0.974
0.817
0.998
0.991
0.952
IRlimit
Max CV
(%)
1
1
1
1
1
1
1
1
1
1
None
None
34
893
1766
None
None
None
None
None
Best
weight
(lb)
NSA
Mean
weight
(lb)
Worst
weight
(lb)
SD
Max
ISD
ISDlimit
Fig. 9 illustrates the convergence history of EJA and other algorithms to the optimum design.
In this example, number of structural analyses required by EJA in order to find the optimum
design is 16306. Despite the fact that the convergence rate of EJA is slightly slower than the
others, it reaches the best feasible design. It can be explained as comparing the convergence
behavior of the unfeasible designs with feasible ones is pointless.
Figure 9. Comparison of convergence curves for the three-bay twenty-four story frame
5.5. 224 Member Braced Frame
The last comparison example is 224 member braced frame which was firstly optimized
by Hasancebi et. al. [16] using seven different algorithms: SA, ESs, TS, ACO, PSO, SGA and
HS.
(a) Side view
(b) Plan view
Figure 10. 224 member braced frame
Fig 10. demonstrates side and plan views of the example which represents one of the interior
frameworks of the structure along the short side direction. The height is 276 ft and the frame
has three bays that each one has a span of 30 ft. The non-swaying concept is provided by bracing
the frame with X-type system placed within the middle bays. The further trusses are also placed
at the twelfth and the top story in order to increase lateral rigidity and hence the large
displacements are prevented. The members are collected in 32 groups that specified as interior
columns, exterior columns, beams and diagonals of successive three stories as shown in Fig
10.(a). The material properties of the steel are as follows: modulus of elasticity (E) is 29000 ksi
and yield strength (Fy) is 36 ksi. The column members are selected from the wide-flange profile
list consisting of 297 sections while beams and diagonals are restricted to be selected from
discrete sets of 171 and 147 economical sections classified according to the properties of area,
inertia and radii of gyration. The single loading condition is the combination of the gravity, live,
snow and wind load that are calculated as to ASCE7-05 [33]: a design dead load of 60.13 lb/ft2,
a design live load of lb/ft2 a ground snow load of 25 lb/ft2 and a basic wind speed of 91 mph
resulting in a uniformly distributed gravity load of 1001.62 lb/ft on top story beams, and of
1453.72 lb/ft on other story beams. Wind loads applying at each floor level on windward and
leeward faces of the frame are tabulated in Table 5. The strength, stability and displacement
constraints are considered according to the provisions of ASD-AISC [28].
Table 5. Wind loads acting on 224-member braced frame
Windward
Leeward
Floor
(kips)
(kips)
1
1.69
2.454
2
1.933
2.454
3
2.17
2.454
4
2.356
2.454
5
2.512
2.454
6
2.646
2.454
7
2.765
2.454
8
2.872
2.454
9
2.971
2.454
10
3.062
2.454
11
3.146
2.454
12
3.225
2.454
13
3.3
2.454
14
3.371
2.454
15
3.438
2.454
16
3.502
2.454
17
3.563
2.454
18
3.621
2.454
19
3.678
2.454
20
3.732
2.454
21
3.784
2.454
22
3.835
2.454
23
3.884
2.454
24
1.966
1.227
The optimum results of 224 member braced frame using JA, EJA and other methods are
tabulated in Table 6. The EJA obtained the best design weighing of 219845.9 lb that is %11
lighter than the SA which has the best design among seven methods in the study of Hasancebi
[16]. In addition, EJA obtained %7.2 lighter design with 1491 less number of structural analyses
and with 7206 less standard deviation value than JA.
Table 6. Comparison of optimum designs for 224 member braced frame
Sizing
variables
Optimal cross section (area. in2)
SA
[16]
ESs
[16]
TS
[16]
ACO
[16]
PSO
[16]
SGA
[16]
HS
[16]
JA
EJA
This Study This Study
1
W14×109
W14×109
W12×120
W14×120
W27×146
W18×143
W24×146
W14×145
W14×120
2
W40×277
W40×277
W36×280
W40×268
W36×260
W40×328
W14×233
W14×257
W36×280
3
W8×40
W10×39
W8×40
W10×45
W10×39
W10×39
W10×49
W10×33
W8×24
4
W16×40
W16×40
W16×45
W16×40
W16×40
W16×40
W16×40
W18×40
W16×36
5
W14×99
W30×108
W18×106
W33×118
W18×130
W18×119
W14×109
W18×119
W14×99
6
W12×190
W12×210
W30×191
W40×221
W14×176
W21×201
W21×201
W30×191
W24×192
7
W10×39
W8×35
W8×35
W8×35
W8×35
W8×35
W10×39
W8×24
W6×20
8
W16×45
W14×43
W16×45
W14×43
W16×40
W16×40
W16×40
W16×40
W21×44
9
W14×90
W27×94
W18×97
W14×90
W21×101
W14×99
W21×101
W10×100
W14×90
10
W14×145
W14×145
W40×167
W33×152
W21×182
W12×152
W14×132
W36×150
W36×150
11
W8×31
W8×35
W10×33
W14×43
W8×401
W10×39
W12×45
W6×20
W8×24
12
W16×45
W14×43
W16×45
W18×50
W16×45
W16×45
W16×50
W21×44
W18×40
13
W30×90
W30×90
W27×94
W30×90
W21×101
W30×99
W21×147
W24×104
W14×90
14
W27×114
W30×116
W10×112
W27×129
W10×112
W30×235
W21×147
W27×129
W10×112
15
W8×40
W8×40
W10×39
W8×35
W8×35
W8×31
W14×61
W8×24
W10×22
16
W18×50
W18×50
W16×50
W18×60
W18×50
W24×76
W24×68
W16×40
W21×44
17
W10×68
W21×73
W24×76
W21×83
W12×87
W16×67
W14×82
W14×82
W24×94
18
W24×104
W24×104
W18×97
W24×104
W18×86
W14×99
W14×109
W30×124
W27×102
19
W8×31
W8×31
W8×31
W8×31
W8×31
W10×33
W10×33
W5×19
W6×15
20
W16×45
W14×43
W16×50
W16×40
W16×40
W14×43
W16×40
W18×46
W21×44
21
W14×53
W24×76
W14×53
W21×62
W24×84
W10×60
W18×71
W10×68
W16×57
22
W12×72
W8×31
W14×68
W21×73
W12×53
W14×74
W14×90
W12×72
W12×65
23
W8×31
W8×31
W8×31
W8×31
W8×31
W8×35
W8×48
W8×21
W6×15
24
W16×40
W16×40
W16×40
W16×40
W24×68
W18×50
W16×45
W14×38
W18×40
25
W16×40
W16×40
W14×43
W16×67
W21×68
W16×45
W14×74
W16×57
W18×50
26
W10×54
W10×49
W12×45
W12×53
W10×39
W12×53
W27×129
W10×54
W16×57
27
W8×31
W8×31
W8×31
W10×33
W8×31
W8×40
W12×40
W6×20
W6×15
28
W16×40
W16×40
W16×40
W16×40
W16×40
W16×45
W16×40
W18×35
W18×35
29
W8×31
W8×31
W8×35
W14×53
W24×76
W10×60
W21×83
W8×40
W12×35
30
W8×35
W8×35
W10×33
W12×45
W14×61
W16×67
W14×61
W10×100
W8×28
31
W8×31
W8×31
W10×33
W10×33
W8×31
W8×31
W10×49
W6×20
W6×15
32
W14×43
W14×43
W18×55
W18×55
W24×68
W24×68
W18×65
W16×31
W18×35
247053.34
248226.5
255486.55
264471.82
270204.24
285261.62
300092.65
236959.5
219845.9
50000
50000
50000
50000
50000
50000
50000
29427
27932
Mean
weight (lb)
N/A
N/A
N/A
N/A
N/A
N/A
N/A
257403.46
236434.7
Worst
weight (lb)
N/A
N/A
N/A
N/A
N/A
N/A
N/A
286639.9
258593.1
SD
N/A
N/A
N/A
N/A
N/A
N/A
N/A
21222.31
14016.71
Max ISD
0.00166
0.001715
0.001625
0.001582
0.001556
0.001503
0.001436
0.001786
0.001831
ISDlimit
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
0.00333
Best
weight
(kg)
NSA
Max IR
0.9979
1.81
0.9885
0.9647
0.9982
0.9726
0.9887
0.9947
0.9986
IRlimit
Max CV
(%)
1
1
1
1
1
1
1
1
1
None
81
None
None
None
None
None
None
None
The convergence history curves of all algorithms are plotted in Fig. 9. It should be noted
that the number of structural analyses for the EJA is almost half of other methods and this is a
shred of strict evidence that EJA has a great convergence performance compared to all methods
given in Fig 9. Although JA does not converge as fast as ESs [16] and TS [16] methods, it
eventually obtains lighter designs than both.
Figure 11. Comparison of convergence curves for the 224-member braced frame
6. Conclusions
JA is one of the most efficient optimization algorithms using only one simple modification
equation without specific parameters based on approaching the best solution meanwhile getting
away from the worst ones. In this study, the standard formulation of JA was modified in order
to enhance the exploration capacity with less structural analyses and avoid trapping in local
optima also applying a dynamic termination criterion for preventing useless iterations which do
not affect the optimization process.
Frame structures are relatively more complex optimization problems since involving more
parameters and formulations unlike truss structures. In order to tackle the complexity and attain
better solutions, the enhanced Jaya algorithm (EJA) was proposed for the weight minimization
of planar steel frames by size tuning of structural members. To test the validity of the algorithm,
five planar steel frame structures were optimized and the results were compared with those of
optimization methods. Remarkably, EJA found the best design compared to all methods in
every single example and strictly satisfied the constraints. The statistical results indicate that
EJA has better performance than the standard JA and other algorithms in terms of robustness,
convergence speed and feasibility.
The competitive performance of EJA allows the researchers to apply the method to other
engineering problems. Further studies are currently conducted for using EJA in the optimization
of large-scale 3D steel frames with a large number of design variables.
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