1 INTRODUCTION

Optimization is among the most interesting and attention engaging topics for engineers
in various fields. In other words, we can apply the optimization techniques to solve
several engineering problems. Some of these utilizations are mentioned in ^{Yang (2010)} as instances. Choosing and assigning
specific sections to structural elements, in addition to satisfying the design criteria
and minimizing the weight of the structure, is an aim which can be achieved by applying
the optimization techniques. In general, techniques applied in structural optimization
can be categorized into classical and heuristic search methods. Classical optimization
methods include mathematical programming and optimality criteria (^{Kaveh and Talatahari, 2010}). There are numerous applications of
these optimization techniques in the literature, but heuristic methods have been among
interest in recent decades because of their advantages in comparison with other two
methods.

Optimization using the uniform deformation theory (UDT) is another approach which has
been proposed and applied to optimum design of the structures in recent years. This
method is based upon the original work of ^{Karami
Mohammadi (2001)} and quite different from other mentioned methods in the field
of optimization. It's formed based on the concept of structural performance and uniform
distribution of deformation demands in the structure subjected to the seismic
excitation. The aim of this methodology is to assign specific sections to members such
that all of the members can reach their allowable deformation capacity during the
earthquake. It has been shown that in this status, the weight of the structure is
minimum, and therefore, the design of the structure is optimized (^{Karami Mohammadi et al., 2004}). According to the basis of this
approach, UDT can be assumed as a method of Performance-Based Design Optimization
(PBDO).

In recent years, researchers have used this approach for the optimization of different
structures such as work of ^{Rahemi et al. (2007)},
^{Moghaddam et al. (2009)}, ^{Hajirasouliha et al. (2011)} and ^{Karami
Mohammadi and Sharghi (2014)}. However, the optimality of the designs has been
proved only by comparing them with the conventional force-based designs. The results of
this method have not yet been compared with the results of other optimization
techniques.

This paper presents an algorithm to PBDO of steel moment frames using the concept of the UDT. The algorithm consists of two phases. In the first phase, to enhance the rate of convergence, the search space of design variables is assumed to be continuous. In this phase, only the deformation-controlled elements may vary. In the second phase, to reach a practical design, discrete cross sections in the neighborhood of the results gained in the previous phase are selected for each element. Acceptance criteria for both deformation and forced-controlled elements are controlled to be satisfied. In order to confirm the suitability of the proposed method, the results of the PBDO of the two steel moment frames are compared with the results of three well-known metaheuristics including Genetic Algorithm (GA), Ant Colony Optimization (ACO) and Particle Swarm Optimization (PSO). Comparison study represents high speed of the proposed algorithm of UDT to achieve the optimal solution in compared with metaheuristics.

2 PERFORMANCE-BASED DESIGN OPTIMIZATION

Performance-based design (PBD) procedure is relatively a new concept for seismic design
of structures which has been introduced in early 1990s as an alternative to the current
strength-based design methods (^{FEMA, 2006}). The
growing acceptability of the performance-based design approach is reflected by
investigations related to seismic rehabilitation of existing buildings which have been
published by Federal Emergency Management Agency (FEMA), the Structural Engineers
Association of California (SEAOC), the Applied Technology Council (ATC), California
Universities for Research in Earthquake Engineering (CUREE) and SAC (a joint venture of
SEAOC, ATC and CUREE). Principles and concepts governing these guidelines for seismic
rehabilitation could also be used to construct new buildings in the form of
performance-based design (^{Gong, 2003}).

This design approach includes some procedures by which a structure designed such a way
that its performance guarantees a predefined objective performance under seismic
loading. Each performance objective is a combination of structural and non-structural
components performance levels which is defined as overall structural performance level
and in fact it is an expression of acceptable damages and losses in a specific hazard
level. In PBD codes such as ^{ASCE 41-06 (2007)},
performance levels for a building consists operational (OP), immediate occupancy (IO),
life safety (LS) and collapse prevention (CP). Additionally the considered hazard levels
are defined based on the probability of exceedance in a specific period (often 50
years). Conventional assumption is that OP, IO, LS and CP performance level correspond
with 50%/50 year, 20%/50 year, 10%/50 year and 2%/50 years, respectively.

In order to evaluate the seismic demands at different performance levels, according to
^{ASCE 41-06 (2007)}, linear procedures (Linear
Static Procedure and Linear Dynamic Procedure, LSP and LDP) and the nonlinear procedures
(Nonlinear Static Procedure and Nonlinear Dynamic Procedure, NSP and NDP) by considering
defined limitations for each of them can be used. In this research, pushover analysis
(or NSP) is considered to determine the nonlinear response of the structures. Pushover
analysis because of simplicities in comparison with NDP is widely used to predict
nonlinear response of the structures. In this method the structure shall be subjected to
monotonically increasing lateral loads representing inertia forces in an earthquake
until a target displacement is exceeded. The target displacement is intended to
represent the maximum displacement likely to be experienced during the design
earthquake. Based on ^{ASCE 41-06 (2007)} the target
displacement (δ*
_{t}
*) is defined as:

where *C*
_{0} relates spectral displacement to the building roof displacement,
*C*
_{1} relates expected maximum inelastic displacements to displacements
calculated for linear elastic response, *C*
_{2} represents the effect of hysteresis shape on the maximum displacement
response, *T*
_{e} is effective fundamental period of the building, *S*
_{a} is the response spectrum acceleration corresponding to the
*T*
_{e} and *g* is gravity acceleration.

Simultaneously with introduction of PBD and develop of related guidelines in the 1990s, the subject of optimization in PBD framework, performance-based design optimization (PBDO), was also considered by researchers. In general a structural optimization problem can be formulated as follows:

where X is a set of design variables (e.g. cross-sectional area of structural element
groups); *ng* is the number of design variables; *D*
*
_{i}
* represents a set of allowable values for the design variable

*i*; W (

*X*)is the weight of the structure; γ

*,*

_{i}*x*

*and*

_{i}*L*

*represent the weight per unit volume, cross-sectional area and the length of element*

_{i}*i*,respectively;

*nm*is the number of elements;

*g*

*(*

_{j}*X*) determines the design constraints and

*nc*is the number of constraints.

In order to determine *g*
*
_{j}
* (

*X*) which indicates performance criteria in a problem of PBDO, the demand capacity ratio (DCR) of structural element groups should be calculated. According to code specifications, the DCR of element group

*i*can be expressed as

where is maximum
rotation of the plastic hinge in element group *i*, and is the allowable rotation of
element group *i* corresponding to a specific performance level which can
be determined from ^{ASCE 41-06 (2007)}.
and
are the maximum
axial load and bending moment for element group *i*, respectively;
and
are the allowable
axial load and bending moment for element group *i* which shall be
calculated in accordance with ^{AISC 360-10 (2010)}.
Hence, constraint for each structural element group is evaluated as follows:

Also rotation is replaced here by curvature, based on the assumption that DCR of rotation and curvature are almost the same.

3 METAHEURISTIC ALGORITHMS

The metaheuristics are the optimization techniques developed in the last two decades
(^{Kaveh and Shojaee, 2007}). These techniques
are usually random and iterative procedures and are involved with discrete variable
designs, although these methods have also been used for the optimization problems with
continuous variables. The fundamental of these algorithms is normally dependent on their
similarity to natural and social processes. Some of these methods include Genetic
Algorithms (GAs), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO),
Simulated Annealing (SA), Harmony Search (HS), Charged System Search (CSS), Imperialist
Competitive Algorithm (ICA) and Big Bang-Big Crunch (BB-BC).

In this paper, GA, PSO and ACO which are among the most popular algorithms in the field of optimization and numerous successful applications of them have been reported in the literature are used to PBDO of the structures. Basic concepts of these algorithms are briefly described below.

3.1 Genetic Algorithm

Genetic algorithm (GA) was first introduced by ^{Holland (1975)}. It simulates the natural evolutionary process to generate
better or fittest species to survive the environment. A genetic algorithm operates on
a population of individuals. Each individual as a design (solution) includes a set of
values which are assigned to design variables and are shown with a string of binary
digits. Algorithm makes a population as initial designs randomly; then the
combination of individuals (designs) occurs to create a population for next
generation. In this procedure in which natural evolution of living organisms is
mimicked, the combination of individuals happens based on a selection procedure. Each
individual is first evaluated and a fitness value corresponding to the objective
function is allocated to it. Next, individuals with high fitnesses are selected for
reproduction. An individual with high fitness has several chances for pairing in
reproduction phase. Hence, a probability is allocated to each individual in the
population based on its fitness to be selected as a parent. Next generation are
developed from selecting pairs of parent and the application of explorative operators
such as mutation and crossover. Crossover is a process in which selected parent
string is divided into parts and some of these parts are exchanged with corresponding
parts of another parent string. Mutation process enables the children to have
characteristics that don't exist in both parent strings. Without this operator, some
regions of search space may never be discovered.

Therefore by using three basic operators of GA including selection, crossover and
mutation, next generation of population which has better fit individuals in
comparison with previous generation will be created. The main philosophy of a GA is
that at every time after start of the process, by combining the more fit individuals,
the average fitness of the population should be increased and the algorithm is
converged to an optimal point. More details of this method can be seen in work of
^{Camp et al. (1998)} and ^{Erbatur et al. (2000)}.

So far the standard GA (SGA) and its improved versions widely adopted by different
researchers in various engineering optimization problems, some of these applications
can be found in the studies conducted by ^{Farhat et al
(2009)}, ^{Kociecki and Adeli
(2013)}.

3.2 Ant Colony Optimization

Ant Colony Optimization (ACO) was first proposed by ^{Colorni et al. (1991)} and ^{Dorigo et al.
(1991)} as a multi-agent approach to solve different combinatorial
optimization problems like the travelling salesman problem (TSP). ACO is inspired by
the behavior of real ants which are able to seek the shortest path between their
colony and source of food through a complex set of pheromone trails. In ACO
procedure, the shortest path corresponds to the optimum solution for the optimization
problem which is discovered by colony of artificial ants. Each artificial ant
assigning allowable discrete values to design variables represents a solution for the
optimization problem. For each variable, the number of virtual paths which can be
selected by each ant in the colony is equal to the number of discrete values
considered for that variable.

The basic steps of ACO can be explained as follows. At the first iteration an initial value of the pheromone is allocated to each of the paths. This value may be considered as

where τ_{0} is the initial pheromone on all paths, and *W*
*
_{min}
* is the weight of frame resulting from assigning the smallest available
cross-sectional area to each element group.

Then the ant colony including a predefined number of ants is constructed. At the
start of any iteration, each ant is assigned to an element group (*i*
∈ {1, 2, ..., *ng*}) that is considered as the initial point of its
travel. Each ant assigns a section to its corresponding element group using the
selection probabilities of the paths which is determined as follows:

where is the
selection probability of *j*th path for *i*th design
variable by *k*th ant at time *t*, τ*
_{ij}
* (

*t*) is the remaining pheromone trail intensity on the path (i, j), υ

*is the visibility parameter associated with the path (i, j),*

_{ij}*N*

*is the number of available sections in the design database of*

_{i}*i*th element group, α and β are constant parameters which are used to control the relative importance of pheromone trail and visibility, respectively. Visibility parameter in Eq. (6) is calculated as

where *A*
*
_{ij}
* is the

*j*th cross-sectional area for element group

*i*th.

After selecting a path by an ant, pheromone intensity on this path is relatively reduced using local pheromone update equation as follows:

where ξ is the local update parameter between 0 and 1 representing the persistence of pheromone.

Consequently the selection probabilities of the paths using updated values of the
pheromone will be calculated again and the next ant will do its selection. When all
ants did their first choices, they proceed for their next element group
(*i*+1), and whenever an ant's element group is greater than the
number of element group (here greater than *ng*), it will proceed to
element group 1. This process will continue until all ants in the colony assign a
section to all structural element groups. Then the pheromone intensity is updated in
order to increase the pheromone value associated with good or promising paths. The
updating is achieved using global pheromone update equation as follows:

where ρ is a constant between 0 and 1 representing the persistence of pheromone
trails and (1 - ρ) is the evaporation rate between time *t* and
*t* + *ng* (the amount of time required to complete
a cycle); is the
enhanced pheromone amount by the elitist ant which is calculated using following
equation:

In the above equation *W*
^{+} indicates the minimum weight of the structure found by the elitist
ant.

At this point, an iteration of ACO is complete, and a new iteration may be initiated.
More details of the method are explained in work of ^{Camp et al. (2005)} and ^{Kaveh and
Talatahari (2010)}.

The successful application of this algorithm has been already proved by many
researchers in the field of structural optimization. Some applications of the ACO are
mentioned in studies conducted by ^{Hasançebi and
Çarbaş (2011)} and ^{Aydoğdu and Saka
(2012)}.

3.3 Particle Swarm Optimization

The Particle Swarm Optimization (PSO) algorithm was first proposed by ^{Kennedy and Eberhart (1995)}. It's motivated from
the social behavior of bird flocking or fish schooling. PSO algorithm includes a
population of individuals which move in search space and each individual possess a
specific speed, which operates as an operator to obtain a new set of individuals.
Individuals, who are called particles, adjust their movements based on their own
experience and the experience gained by the population (^{Kaveh and Talatahari, 2007}).

Each particle of swarm represents one solution to the optimization problem and its
position updated based on the best position obtained by the particle itself and also
by the best position of the swarm in each repetition. Numerically, the position
*x* of a particle *i* at iteration
*k* + 1 is updated as Eq.(11)

where is the
corresponding updated velocity vector and Δ*t* is the value of time
step (usually assumed to be one).

The velocity vector for each particle in each step is expressed as follows:

where is the
velocity vector at iteration *k*, and are the best position for the
particle *i* and the global best position in the swarm up to iteration
*k*, respectively, *r*
_{1} and *r*
_{2} are two random numbers in the interval [0,1]. The remaining terms are
the configuration parameters which possess an important role in PSO convergence
behavior. So that the coefficients *c*
_{1} (cognitive parameter) and *c*
_{2} (social parameter) represent degree of confidence in the best solution
found by each individual particle and by the swarm as a whole, respectively. The
final term ω, is the inertia weight which is employed to control the exploration
abilities of the swarm and in general scales the current velocity value affecting the
updated velocity vector. It is proved that to guarantee the convergence of PSO, these
coefficients should satisfy the following conditions:

In this research, to update inertial weight in each repetition, a linear reduction technique is used which is defined as Eq.(14).

In the above equation ω*
_{max}
* and ω

*are initial and final values of inertia weight and*

_{min}*k*

*is the maximum number of iterations. More details of the method are presented in work of*

_{max}^{Perez and Behdinan (2007)}.

In field of structural optimization many successful applications of PSO have been
published by various authors. Some of these applications can be found in work of
^{Luh and Lin (2011)}, ^{Kaveh and Zolghadr (2014)}.

4 OPTIMIZATION BASED ON UNIFORM DIFORMATION THEORY (UDT)

Studies and investigations done by different researches in several fields such as the effect of dynamic nature of seismic forces in the response of the structures, lateral load distribution patterns and their influence on the deformation demands, and the optimum distribution patterns of shear strength and stiffness in structures led to introduce a new concept called uniform deformation theory.

Initial algorithm of this method first proposed by ^{Karami
Mohammadi (2001)} as an iterative procedure to determine the optimum strength
distribution pattern for a shear building model subjected to a given earthquake. Based
on this algorithm, then ^{Moghaddam and Hajirasouliha
(2004)} proposed an approach called uniform deformation algorithm and used it
for optimum seismic design of a shear and truss-like structure. In studies carried out
by these researchers, the ductility of the story and the ductility of the members are
assumed as the demand parameters to control the performance of the shear and truss-like
structure, respectively.

Based on this theory, inefficient material is gradually shifted from the strong to weak
areas leads to a uniform deformation (ductility) state at the end of repetitive process.
It has been shown that in this status the seismic performance of the structure is
optimized. Although the base of this theory and proposed algorithm is to attain a
uniform state of deformation in the whole structure and studies on this theory and its
application in the field of structural optimization has been rests on the same base, but
the allowable limit of deformation values defined in (PBD) codes such as ^{ASCE 41-06 (2007)} is not constant for all of
structural members. On the other hand, in these codes, some actions of structural
members may be controlled by deformation and some controlled by force (see Eq.(3)). For example the flexural actions of
beams shall be considered deformation-controlled while the flexural loading of columns
depending on the amount of the axial load may be controlled by force or deformation.
Therefore, by considering the acceptance criteria of PBD codes, it is not possible to
reach a uniform deformation state in the whole structure and then using the expression
of uniform deformation algorithm loses its meaning somewhat.

According to the basic concepts of this theory and the algorithm which has been proposed previously, we tried to present a method to let members reach their allowable deformation or strength capacity. The proposed method consists of two phases. In the first phase of the search, to enhance the rate of convergence, the search space of design variables is assumed to be continuous. Therefore, in any iteration plastic section modulus is modified and other cross-sectional properties can be determined accordingly by linear interpolation. Additionally in this phase of search, only the deformation-controlled elements may vary, thus DCR of force-controlled members is assumed to be one.

First phase of the search is done as follows:

1. For the initial design, the cross-sectional area of all members is supposed their maximum available. Therefore the assumed weight of the structure is the maximum in the initial step.

2. The structure is analyzed and the DCRs are calculated for each structural member group from Eq.(3).

3. The coefficient of variation (COV) of groups' DCRs is determined using following equation:

where *DCR*
*
_{ave}
* is the average of DCR.

4. If the termination criteria are satisfied the optimization process will be stopped in the first phase. Otherwise the process continues. The termination criteria can be expressed as follows.

COV reduced to the desired value (e.g. less than about 10%), while *DCR*
*
_{ave}
* is greater than the predefined value (e.g. greater than about 70%) or the
variation of weight is small enough (e.g. less than about 0.1%).

where [*Z*
*
_{i}
*]

*and [Z*

_{k}*]*

_{i}*are the plastic section modulus of element group*

_{k + 1}*i*at iteration

*k*and

*k*+ 1, respectively,

*c*

*is convergence coefficient which will be calculated for each member group using Eq.(17).*

_{i}

In the above equation ψ is a constant between 0 and 1 which is taken 0.3 in this research.

By using the new property of plastic section modulus, other cross-sectional properties can be calculated.

Second phase of the search is a two-step process. First for each structural member groups, the nearest discrete section to the imaginary section achieved in the first phase is identified and selected. In the second step the structure is analyzed again and the DCR of each group is calculated. In cases where this ratio is greater than one for a group, it is assigned to the stronger section. In this phase, acceptance criteria for both deformation and forced controlled elements are supposed to be satisfied.

5 NUMERICAL EXAMPLES

In this section two baseline steel moment frames, four-bay three-story and five-bay
nine-story, using described methods are optimized. These frames are adopted from model
buildings investigated in the SAC steel project (^{FEMA,
2000}) located in the Seattle area and their general specifications, including
geometry, loading and material properties selected accordingly. The nine-story model is
slightly modified for this study. The buildings are assumed to be located on a soil type
C. The modulus of elasticity and yield stress of steel material are 200 GPa and 345 MPa,
respectively, and the strain hardening slope is equal to 3% of the elastic modulus. IPB
sections are chosen for columns, while IPE sections and eight sections of plate girders
(PG1-PG8) with the predefined properties are considered for beams. Weight per unit
length of plate girder sections is presented in the following Table 1.

According to the code recommendation, lateral load pattern is assumed based on the first
mode shape of the frame (^{ASCE, 2007}). Also the
following gravity load is considered for combination with the seismic loads:

where *Q*
*
_{D}
* and

*Q*

*are dead and live loads, respectively.*

_{L}In order to calculate the target displacement and perform the pushover analysis, design
acceleration spectrum is considered in accordance with ^{ASCE 7-10 (2010)} and can be expressed as follows:

where *S*
*
_{DS}
* and

*S*

*are design spectral response acceleration parameters at short periods and period of 1 second, respectively, T*

_{D1}*is the long-period transition period which may be determined according to the site in which the structure is located as 6s. Also T*

_{L}_{0}and T

*will be calculated using the following equations:*

_{S}

In this study design is performed based on the life safety (LS) performance level and hazard level corresponding to 10%/50 year, thus the design spectral response acceleration parameters are assumed to be 2/3 values of these parameters in the maximum considered earthquake (MCE), and can be determined as follows:

where *S*
*
_{MS}
* and

*S*

*are the MCE spectral response acceleration parameters at short periods and period of 1 second, respectively. These parameters are defined as*

_{M1}

In the above equation *S*
*
_{S}
* and

*S*

_{1}are the mapped MCE spectral response acceleration parameters at short periods and period of 1 second, respectively. Based on the presented maps in

^{ASCE 7-10 (2010)}, these parameters for the Seattle area may be determined as 1.360 g and 0.527 g, respectively. Additionally

*F*

*and*

_{a}*F*

*are the site coefficients which can be determined based on the soil type and values of*

_{v}*S*

*and*

_{S}*S*

_{1}as 1 and 1.3, respectively.

For both presented examples, the parameters of the metaheuristics are taken based on the
ranges of these values and also by considering the problem conditions to achieve the
best results. In this way, the GA, ACO and PSO parameters are adopted based on works of
^{Kaveh et al. (2010)}, ^{Kaveh and Talatahari (2010)} and ^{Kaveh and Talatahari (2008)}, respectively. These values are presented in
Table 2.

Metaheuistic algorithm | Values of parameter set |

GA | crossover fraction = 0.8, mutation fraction = 0.2 |

ACO | α = 1, β = 0.4, ξ = 0.25, ρ = 0.2 |

PSO | c_{1} = c_{2} = 0.8, ω_{min} =
0.4, ω_{max} = 0.9 |

Moreover in all three methods in order to handle the design constraints, an exterior penalty function is used. In this case, the aim of the optimization is redefined by using a penalty function as

where *W*
*
_{penalized}
* (

*X*) is the structural penalized weight (objective function) and

*f*

*(*

_{penalty}*X*) is the penalty function which can be expressed as follows:

In the above equation *v* specifies the total violations of design
constraints. Also the constants ε_{1} and ε_{2} are selected considering
the exploration and the exploitation rate of search space (^{Kaveh and Talatahari, 2010}). In this paper ε_{1} and
ε_{2} are taken as 1 and 2, respectively for all three methods.

Due to random nature of metaheuristic algorithms, the optimization problem is solved
independently five times with each method. Five runs of each algorithm seem to be
adequate in order to reach the acceptable design and also close to the best design which
probably is achieved in an infinite number of runs (^{Hasançebi et al., 2010}). In contrast with metaheuristics, proposed algorithm
based on the uniform deformation theory is a deterministic method and then the problem
is solved once using this method.

5.1 Four-bay three-story steel frame

The geometry and grouping details of the four-bay three-story frame are shown in
Figure 1. The 27 members of the frame are
classified into five groups, as indicated in the figure. The dead load of
Q_{D} = 21 kN/m is applied to the first and second story beams, while the
dead load of Q_{D} = 18.2 kN/m is applied to the roof beams. Also the live
load of Q_{L }= 4.4 kN/m is considered to all stories beams. The seismic
weights for the structure are considered as 4689 kN for the first and second stories,
and 5073 kN for the roof story. For all three metaheuristics, the population size and
the total number of generations are considered as 20 and 50, respectively.

The results of optimization including the best, the worst and the average weight obtained for the frame and also sections achieved for each of the structural element group in the best run is provided in Table 3. The weight of frame obtained using the proposed method, based on UDT, is equal to 212.98 kN, which is 4.33% lighter than the best result of GA and also 6.67% and 5.58% heavier than the best results of ACO and PSO methods, respectively. In addition the result of proposed method is lighter than the worst results found by all methods. A comparison between the average results of five runs for each of the metaheuristic algorithm indicates that the obtained result using UDT is 12.13% and 1.21% lighter than the average results of GA and ACO, respectively, and it is also 1.41% heavier than the average results of PSO. The convergence history of the proposed method is compared with the metaheuristic algorithms in Figures 2 and 3 for the best and average runs, respectively. As can be seen, the convergence rate in the proposed method is much higher than the metaheuristic algorithms. The proposed method needed 52 analyses for convergence which is lower than 552, 508 and 268 analyses required by ACO, PSO and GA on average, respectively.

Element Group | GA | ACO | PSO | Present Word (Based on UDT) |

1 | HE500B | HE900B | HE500B | HE220B |

2 | HE650B | HE220B | HE360B | HE650B |

3 | IPE500 | IPE400 | IPE600 | IPE600 |

4 | IPE600 | PG3 | IPE600 | PG1 |

5 | IPE400 | IPE360 | IPE360 | IPE400 |

Best weight (kN) | 222.61 | 199.66 | 201.72 | 212.98 |

Average weight (kN) | 242.39 | 215.58 | 210.01 | - |

Worst weight (kN) | 263.80 | 231.11 | 218.55 | - |

Average no. of analyses | 268 | 552 | 508 | 52 |

Figure 4 shows the DCR of element groups in the optimum designs for all methods. It is apparent from the figure that all of the DCRs are lower than one.

5.1 Five-bay nine-story steel frame

Figure 5 shows the geometry and grouping details
of the five-bay nine-story frame. The frame is composed of 99 members have been
classified into nine groups, as illustrated in Figure
5. The dead load of Q_{D} = 21 kN/m is applied to beams in the
first to the eighth stories, while the dead load of Q_{D} = 18.2 kN/m is
applied to the roof beams. Also the live load of Q_{L }= 4.4 kN/m is
considered to all stories beams. The seismic weights for the structure are considered
as 4940 kN for the first story, 4855 kN for the second to eighth stories, and 5230 kN
for the roof story. For this example in all three metaheuristics, the population size
and the total number of generations are considered as 50 and 80, respectively.

Table 4 lists the designs developed by the metaheuristic algorithms in the best run of them, and also by the present work. Proposed design based on the UDT results frame weight of 814.24 kN, which is 16.19%, 7.04% and 0.56% lighter than best design of GA, PSO and ACO, respectively. Propose algorithm reaches to its best design in 37 analyses which is much lower than 2480, 1660 and 1600 analyses required by ACO, PSO and GA on average, respectively. The best and the average convergence history of the metaheuristics are compared with proposed algorithm in Figures 6 and 7, respectively. As can be seen, the optimization based on the UDT has high convergence rate compared to the metaheuristics.

Element Group | GA | ACO | PSO | Present Work (Based on UDT) |

1 | HE800B | HE600B | HE650B | HE600B |

2 | HE450B | HE450B | HE450B | HE400B |

3 | HE900B | HE360B | HE360B | HE340B |

4 | HE400B | HE280B | HE280B | HE280B |

5 | IPE600 | PG1 | PG1 | PG1 |

6 | PG2 | IPE600 | PG1 | PG1 |

7 | IPE550 | IPE600 | IPE600 | IPE600 |

8 | IPE600 | IPE500 | IPE600 | IPE500 |

9 | IPE330 | IPE330 | IPE450 | IPE330 |

Best weight (kN) | 971.53 | 809.73 | 875.95 | 814.24 |

Average weight (kN) | 1072.30 | 888.07 | 876.20 | - |

Worst weight (kN) | 1207.08 | 999.06 | 876.36 | - |

Average no. of analyses | 1600 | 2480 | 1660 | 37 |

DCRs of element groups for nine-story frame are shown in Figure 8. As indicated in the figure, distribution of DCRs is more uniform (COV of DCRs is smaller) for the ACO and the proposed algorithm in comparison with GA and PSO. Consequently, the designs of these methods are lighter than the design of GA and PSO.

6 CONCLUSIONS

This paper studied the PBDO of a three and nine-story steel moment frame using three metaheuristic algorithms including GA, ACO and PSO. Furthermore the results are compared with a proposed method which is based on the UDT. Based on the criteria of PBD codes, presented deformation capacity for different members of the structure is not equal and consequently forming a uniform state of deformation in the whole structure is not possible. For that reason in the proposed method, the COV of DCR approached to zero instead of COV for deformation of the structural element groups, i.e. almost uniform state of damage is formed in the structure. Results demonstrate that the proposed algorithm has high speed to reach acceptable solution in comparison with results of three metaheuristics. Efficiency of the optimization based on UDT is more obvious in design of nine-story frame, where with the growth of the problem size, the required number of population (or number of analyses) in the metaheuristics to reach the optimum design is increased. In addition, unlike the UDT method, metaheuristic algorithms are non-deterministic which are required to solve problem several times and this also increase the number of analyses needed by the metaheuristics to reach the optimum design.