Applications of Metaheuristic Optimization PDF
Applications of Metaheuristic Optimization PDF
Applications of Metaheuristic Optimization PDF
Kaveh
Applications of
Metaheuristic
Optimization
Algorithms in
Civil Engineering
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Applications of Metaheuristic Optimization
Algorithms in Civil Engineering
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A. Kaveh
Applications of Metaheuristic
Optimization Algorithms in
Civil Engineering
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A. Kaveh
Iran University of Science and Technology
Centre of Excellence for Fundamental Studies
in Structural Engineering
Tehran, Iran
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Preface
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vi Preface
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Preface vii
Every effort has been made to render the book error free. However, the author
would appreciate any remaining errors being brought to his attention through his
e-mail address: alikaveh@iust.ac.ir.
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Metaheuristic Algorithms for Optimization . . . . . . . . . . . . . . . 1
1.2 Optimization in Civil Engineering and Goals of the
Present Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Organization of the Present Book . . . . . . . . . . . . . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Optimum Design of Castellated Beams Using the Tug of War
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Design of Castellated Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Overall Flexural Capacity of the Beam . . . . . . . . . . . . 12
2.2.2 Shear Capacity of the Beam . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Flexural and Buckling Strength of Web Post . . . . . . . . 13
2.2.4 Vierendeel Bending of Upper and Lower Tees . . . . . . . 14
2.2.5 Deflection of Castellated Beam . . . . . . . . . . . . . . . . . . 14
2.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Design of Castellated Beam with Circular Opening . . . 16
2.3.2 Design of Castellated Beam with Hexagonal
Opening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Test Problems and Optimization Results . . . . . . . . . . . . . . . . . 20
2.5.1 Castellated Beam with 4 m Span . . . . . . . . . . . . . . . . . 21
2.5.2 Castellated Beam with 8 m Span . . . . . . . . . . . . . . . . . 23
2.5.3 Castellated Beam with 9 m Span . . . . . . . . . . . . . . . . . 23
2.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ix
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x Contents
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Contents xi
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xii Contents
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Contents xiii
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xiv Contents
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Contents xv
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xvi Contents
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Chapter 1
Introduction
Much has been made of the parallels between engineering and art, and yet a unique
economy of parts and adherence to a plethora of constraints from cost to market
trends, from maintainability to robustness, and from project schedules safely
distinguish engineering design from the arts and engineering projects from art-
works. At the heart of this distinction lies the concept of “optimization” – the
science of choosing design variable values within given constraints such that a
function, e.g., total system cost is minimized, or overall system reliability is
maximized.
While the last three decades has seen an explosion in new methodologies applied
to the problem of optimization, there is also evidence for a resurgence of improved
classical algorithms and a growing number of engineering problems where heuristic
and algorithmic optimization has overtaken and, in some cases, replaced the
engineering graybeards and rule-of-thumb optimization methods.
Some of the most commonly used classical algorithmic optimization techniques
were gradient based and allowed a search of the solution space near a given
parameter point where gradient information about the target function was available
[1, 2]. Gradient-based methods, in general, converge faster and can obtain solutions
of higher accuracy than more modern stochastic approaches. However, the acqui-
sition of gradient information for the target function can be either costly or even
impossible. Moreover, these types of algorithms are only guaranteed to converge to
local minima. Furthermore, a good starting point can be vital for the successful
execution of these methods. In many optimization problems, prohibited zones, side
limits, and non-smooth or non-convex functions need to be taken into consider-
ation, increasing the difficulty of obtaining optimal solutions.
There is a slew of more recently developed optimization methods, known as
metaheuristic algorithms, that are not restricted in the aforementioned manner.
These methods are suitable for global searches over the entire search space due to
their capability of exploring and finding promising regions in the search space with
reasonable computational effort. Ultimately, metaheuristic algorithms tend to per-
form rather well for most optimization problems [3, 4]. This is because these
methods refrain from simplifying or making assumptions about the original prob-
lem. Evidence of this can be seen in their successful application to a vast variety of
fields, such as engineering, physics, chemistry, arts, economics, marketing, genet-
ics, operations research, robotics, social sciences, and politics.
The word heuristic has its origin in the old Greek work heuriskein, which means
the art of discovering new strategies or rules to solve problems. The suffix meta,
also a Greek prefix, has come to mean a higher level of abstraction in the English
language. The term metaheuristic was introduced by Glover in the paper [5] and
denotes a strategy of solving a problem using higher levels of abstractions and to
guide a heuristic search of the solution space.
A heuristic method can be considered as a procedure that is likely to discover a
very good feasible solution, but not necessarily an optimal solution, for a consid-
ered specific problem. In most cases no guarantee is provided for the quality of the
solution obtained, but a well-designed heuristic method usually can provide a
solution that is nearly optimal. The procedure also should be sufficiently efficient
to deal with very large problems. Heuristic methods are often iterative algorithms,
where each iteration involves conducting a search for a new solution that might be
better than the best solution found in a previous iteration. After a reasonable amount
of time when the algorithm is terminated, the solution it provides is the best one
found during all iterations. A metaheuristic is formally defined as an iterative
generation process which guides a subordinate heuristic by combining intelligently
different concepts for exploring (global search) and exploiting (local search) the
search space in order to efficiently find near-optimal solutions [6]. Learning strat-
egies can be employed to add the “intelligence” to such guided search heuristics.
Metaheuristic algorithms have found many applications in different areas of
applied mathematics, engineering, medicine, economics, and other sciences.
Within engineering, these methods are extensively utilized in the design stages of
civil, mechanical, electrical, and industrial projects.
In the area of civil engineering that is the main concern of this book, one tries to
achieve certain objectives in order to optimize weight, construction cost, geometry,
layout, topology, construction time, and computational time satisfying certain
constraints. Since resources, fund, and time are always limited, one has to find
solutions to optimize the usage of these resources.
The main goal of this book is to apply some well established and most recently
developed metaheuristic algorithms to optimization problems in the field of civil
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1.3 Organization of the Present Book 3
The remaining chapters of this book are organized in the following manner:
Chapter 2 introduces the recently developed metaheuristic so-called tug of war
optimization and applies this method to the optimal design of castellated beams.
Two common types of laterally supported castellated beams are considered as
design problems: beams with hexagonal openings and beams with circular open-
ings. In this chapter, castellated beams have been studied for two cases: beams
without filled holes and beams with end-filled holes. Here, tug of war optimization
algorithm is utilized for obtaining the solution of these design problems. For this
purpose, the cost is taken as the objective function, and some benchmark problems
are solved from literature [7].
Chapter 3 presents an integrated metaheuristic-based optimization procedure for
discrete size optimization of straight multi-span steel box girders with the objective
of minimizing the self-weight of girder. The selected metaheuristic algorithm is the
cuckoo search (CS) algorithm. The optimum design of a box girder is characterized
by geometry, serviceability, and ultimate limit states specified by the American
Association of State Highway and Transportation Officials (AASHTO). Size opti-
mization of a practical design example investigates the efficiency of this optimiza-
tion approach and leads to around 15 % of saving in material (Kaveh et al. [8]).
Chapter 4 addresses a new nature-inspired metaheuristic optimization algo-
rithm, called whale optimization algorithm (WOA), and utilizes this algorithm for
size optimization of skeletal structures. This method is inspired by the bubble-net
hunting strategy of humpback whales. WOA simulates hunting behavior with
random or the best search agent to chase the prey and the use of a spiral to simulate
bubble-net attacking mechanism of humpback whales. In this chapter, EWOA is
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4 1 Introduction
also compared with WOA and other metaheuristic methods developed in the
literature using four skeletal structure optimization problems. Numerical results
compare the efficiency of the WOA and EWOA with the latter algorithm being
superior to the standard implementation [9].
Chapter 5 applies the optimum design procedure, based on colliding bodies
optimization (CBO) method and its enhanced version (ECBO), to optimal design of
two commonly used configurations of double-layer grids, and optimum span–depth
ratios are determined. Two ranges of spans as small and large sizes with certain
bays of equal lengths in two directions and different types of element grouping are
considered for each type of square grids. These algorithms obtain minimum weight
grid through appropriate selection of tube sections available in AISC load and
resistance factor design (LRFD). The comparison is aimed in finding the depth at
which each of different configurations shows its advantages. Finally, the effect of
support locations on the weight of the double-layer grids is investigated [10].
Chapter 6 introduces a finite element model based on geometrical nonlinear
analysis of different mechanical systems of large-scale domes consisting of double-
layer domes, suspen-domes, and single-layer domes with rigid connections. The
suspen-dome system is a new structural form that has become a popular structure in
the construction of long-span roof structures. Suspen-dome is a kind of new
prestressed space grid structure which is a spatial prestressed structure and has
complex mechanical characteristics. In this chapter, an optimum geometry and
sizing design is performed using the enhanced colliding bodies optimization algo-
rithm. The length of the strut, the cable initial strain, the cross-sectional area of the
cables, the cross-sectional size of steel elements, and the height of dome are
adopted as design variables for domes, and the minimum volume of each dome is
taken as the objective function. A simple approach is defined to determine the
configurations of the dome structures. The design algorithm obtains minimum
volume domes through appropriate selection of tube sections available in AISC
load and resistance factor design (LRFD). This chapter explores the efficiency of
Lamella suspen-dome with pin-jointed and rigid-jointed connections and compares
them with single-layer Lamella dome and double-layer Lamella dome [11].
Chapter 7 optimizes two single-layer barrel vault frames with different patterns
via the improved magnetic charged system search (IMCSS). In the process of
optimization, contrary to size variables, shape is a continuous variable. In the
case of shape optimization of this type of space structures, since all of the nodal
coordinates as the shape variables are dependent on the height-to-span ratio of the
barrel vault, height is considered as the only shape variable in a constant span of
barrel vault. In comparison, the best height-to-span ratios of barrel vaults under
static loading conditions obtained from CSS, MCSS, and IMCSS algorithms are
approximately close to the value of 0.17 from a comparative study carried out by
Parke. Furthermore, as seen from the results, different patterns of barrel vaults have
different effects on the value of the best height-to-span ratio. Moreover, in com-
parison to CSS and MCSS algorithms, IMCSS found better values for the weight of
the structures with a lower number of analyses [12].
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1.3 Organization of the Present Book 5
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6 1 Introduction
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References 7
References
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8 1 Introduction
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Chapter 2
Optimum Design of Castellated Beams Using
the Tug of War Algorithm
2.1 Introduction
In this chapter, the tug of war algorithm is applied to optimal design of castellated
beams. Two common types of laterally supported castellated beams are considered
as design problems: beams with hexagonal openings and beams with circular
openings. Here, castellated beams have been studied for two cases: beams without
filled holes and beams with end-filled holes. Also, tug of war optimization (TWO)
algorithm is utilized for obtaining the solution of these design problems. For this
purpose, the cost is taken as the objective function, and some benchmark problems
are solved from literature (Kaveh and Shokohi [1]).
Since the 1940s, the manufacturing of structural beams with higher strength and
lower cost has been an asset to engineers in their efforts to design more efficient
steel structures. Due to the limitations on maximum allowable deflections, using
section with heavyweight and high capacity in the design problem cannot always be
utilized to the best advantage. As a result, several new methods have been created
for increasing the stiffness of steel beams without increase in the weight of steel
required. Castellated beam is one of the basic structural elements within the design
of building, like a wide-flange beam (Konstantinos and D’Mello [2]).
A castellated beam is constructed by expanding a standard rolled steel section in
such a way that a predetermined pattern (mostly circular or hexagonal) is cut on
section webs and the rolled section is cut into two halves. The two halves are shifted
and connected together by welding to form a castellated beam. In terms of structural
performance, the operation of splitting and expanding the height of the rolled steel
sections helps to increase the section modulus of the beams.
The main initiative for manufacturing and using such sections is to suppress the
cost of material by applying more efficient cross-sectional shapes made from
standard rolled beam. Web-openings have been used for many years in structural
steel beams in a great variety of applications because of the necessity and economic
advantages. The principal advantage of steel beam castellation process is that
Fig. 2.1 (a) A castellated beam with circular opening. (b) A castellated beam with hexagonal
designer can increase the depth of a beam to raise its strength without adding steel.
The resulting castellated beam is approximately 50 % deeper and much stronger
than the original unaltered beam (Soltani et al. [3], Zaarour and Redwood [4],
Redwood and Demirdjian [5], Sweedan [6], Konstantinos and D’Mello [7]).
In recent years, a great deal of progress has been made in the design of steel
beams with web-openings, and a cellular beam is one of them. A cellular beam is
the modern form of the traditional castellated beam, but with a far wider range of
applications in particular as floor beams. Cellular beams are steel sections with
circular openings that are made by cutting a rolled beam web in a half circular
pattern along its centerline and re-welding the two halves of hot rolled steel sections
as shown in Fig. 2.1. An increase in beam depth provides greater flexural rigidity
and strength to weight ratio.
In practice, in order to support high shear forces close to the connections,
sometimes it becomes necessary to fill certain openings. In cellular beams, this is
achieved by inserting discs made of steel plates and welding from both sides
(Fig. 2.2). The openings are usually filled for one of two reasons:
(i) At positions of higher shear, especially at the ends of a beam or under
concentrated loads
(ii) At incoming connections of secondary beams
It should be noted that for maximum economy infills should be avoided when-
ever possible, even to the extent of increasing the section mass.
In the last two decades, many metaheuristic algorithms have been developed to
help solve optimization problems that were previously difficult or impossible to
solve using mathematical programming algorithms. Metaheuristic algorithms pro-
vide mechanisms to escape from local optima by balancing exploration and exploi-
tation phases, being based either on solution populations or iterated solution paths,
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2.2 Design of Castellated Beams 11
The theory behind the castellated beam is to reduce the weight of the beam and to
improve the stiffness by increasing the moment of inertia resulting from increased
depth without usage of additional material. Due to the presence of holes in the web,
the structural behavior of castellated steel beam is different from that of the
standard beams. At present, there is no prescribed design method due to the
complexity of the behavior of castellated beams and their associated modes of
failure (Soltani et al. [3]). The strength of a beam with different shapes of web-
openings is determined by considering the interaction of bending moment and shear
at the openings. There are many failure modes to be considered in the design of a
beam with web-opening, consisting of lateral-torsional buckling, Vierendeel mech-
anism, flexural mechanism, rupture of welded joints, and web post buckling.
Lateral-torsional buckling may occur in an unrestrained beam. A beam is consid-
ered to be unrestrained when its compression flange is free to displace laterally and
rotate. In this chapter it is assumed that the compression flange of the castellated
beam is restrained by the floor system. Therefore, the overall buckling strength of
the castellated beam is omitted from the design considerations. These modes are
closely associated with beam geometry, shape parameters, type of loading, and
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12 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
This mode of failure can occur when a section is subjected to pure bending. In the span
subjected to pure bending moment, the tee sections above and below the openings
yield in a manner similar to that of a standard webbed beam. Therefore, the maximum
moment under factored dead and imposed loading should not exceed the plastic
moment capacity of the castellated beam (Soltani et al. [3], Erdal et al. [10]).
MU MP ¼ ALT PY H U ð2:1Þ
where ALT is the area of lower tee, PY is the design strength of steel, and HU is the
distance between center of gravities of upper and lower tees.
In the design of castellated beams, two modes of shear failure should be checked.
The first one is the vertical shear capacity, and the upper and lower tees should
undergo that. The vertical shear capacity of the beam is the sum of the shear
capacities of the upper and lower tees. The factored shear force in the beam should
not exceed the following limits:
The second one is the horizontal shear capacity. It is developed in the web post due
to the change in axial forces in the tee section as shown in Fig. 2.3. Web post with
too short mid-depth welded joints may fail prematurely when horizontal shear
exceed the yield strength. The horizontal shear capacity is checked using the
following equations (Soltani et al. [3], Erdal et al. [10]):
where AWUL is the total area of the web-opening and AWP is the minimum area of
web post.
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2.2 Design of Castellated Beams 13
Fig. 2.3 Horizontal shear in the web post of castellated beams. (a) Hexagonal opening. (b)
Circular opening
In this study, it is assumed that the compression flange of the castellated beam is
restrained by the floor system. Thus the overall buckling of the castellated beam is
omitted from the design consideration. The web post flexural and buckling capacity
in a castellated beam is given by Soltani et al. [3] and Erdal et al. [10]):
MMAX
¼ C1 α C2 α2 C3 ð2:4Þ
ME
where MMAX is the maximum allowable web post moment and ME is the web post
capacity at critical section A–A shown in Fig. 2.3. C1, C2, and C3 are constants
obtained by the following expressions
where α ¼ 2d
S
is for hexagonal openings and α ¼ DS0 is for circular openings, also
tw is for hexagonal openings, and β ¼ tw is for circular openings and S is the
β ¼ 2d D0
spacing between the centers of holes, d is the cutting depth of hexagonal opening,
D0 is the hole diameter, and tw is the web thickness.
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14 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
P0 M
þ 1:0 ð2:8Þ
PU M P
where P0 and M are the force and the bending moment on the section, respectively.
PU is equal to the area of critical section PY , and MP is calculated as the plastic
modulus of critical section PY in plastic section or elastic section modulus of
critical section PY for other sections.
The plastic moment capacity of the tee sections in castellated beams with
hexagonal opening is calculated independently. The total of the plastic moment is
equal to the sum of the Vierendeel resistances of the above and below tee sections
(Soltani et al. [3]). The interaction between Vierendeel moment and shear forces
should be checked by the following expression:
where VOMAX and MTP are the maximum shear force and the moment capacity of
tee section, respectively.
Serviceability checks are of high importance in the design, especially in beams with
web-opening where the deflection due to shear forces is significant. The deflection
of a castellated beam under applied load combinations should not exceed span/360.
Methods for calculating the deflection of castellated beam with hexagonal and
circular openings are shown in Raftoyiannis and Ioannidis [12], and Erdal et al.
[10], respectively.
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2.3 Problem Formulation 15
In practice, in order to support high shear forces close to the connection or for
reasons of fire safety, sometimes it becomes necessary to fill certain openings using
steel plates. In this case, the price of plates is added to the total cost. Therefore, the
objective function can be expressed as
where p1, p2, and p3 are the price of the weight of the beam per unit weight, length
of cutting, and welding per unit length, L0 is the initial length of the beam before
castellation process, ρ is the density of steel, Ainitial is the area of the selected
universal beam section, Ahole is the area of a hole, and Lcut and Lweld are the cutting
length and welding length, respectively. The length of cutting is different for
hexagonal and circular web-openings. The dimension of the cutting length is
described by the following equations:
For circular opening,
πD0
Lcut ¼ πD0 NH þ 2eðNH þ 1Þ þ þe ð2:12Þ
2
πD0
Lcut-infill ¼ πD0 NH þ 2eðNH þ 1Þ þ þ e þ 2 Phole ð2:13Þ
2
where NH is the total number of holes, e is the length of horizontal cutting of web,
D0 is the diameter of holes, d is the cutting depth, θ is the cutting angle, and Phole is
the perimeter of hole related to filled opening.
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16 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
Also, the welding length for both of circular and hexagonal openings is deter-
mined by Eqs. (2.16) and (2.17).
Design process of a cellular beam consists of three phases: the selection of a rolled
beam, the selection of a diameter, and the spacing between the center of holes and
total number of holes in the beam as shown in Fig. 2.1 (Erdal et al. [10], Saka [11]).
Hence, the sequence number of the rolled beam section in the standard steel
sections’ tables, the circular holes diameter, and the total number of holes are
taken as design variables in the optimum design problem. This problem is formu-
lated by considering the constraints explained in the previous sections and can be
expressed as the following:
Find an integer design vector fXg ¼ fx1 ; x2 ; x3 gT , where x1 is the sequence
number of the rolled steel profile in the standard sections list, x2 is the sequence
number for the hole diameter which contains various diameter values, and x3 is the
total number of holes for the cellular beam (Erdal et al. [10]). Hence the design
problem can be expressed as follows:
Minimize Eqs. (2.10) and (2.11)
Subjected to
g1 ¼ ð1:08 D0 Þ S 0 ð2:18Þ
g2 ¼ S ð1:60 D0 Þ 0 ð2:19Þ
g3 ¼ ð1:25 D0 Þ H S 0 ð2:20Þ
g4 ¼ H S ð1:75 D0 Þ 0 ð2:21Þ
g5 ¼ M U M P 0 ð2:22Þ
g6 ¼ V MAXSUP PV 0 ð2:23Þ
g7 ¼ V OMAX PVY 0 ð2:24Þ
g8 ¼ V HMAX PVH 0 ð2:25Þ
g9 ¼ MAAMAX MWMAX 0 ð2:26Þ
g10 ¼ V TEE ð0:50 PVY Þ 0 ð2:27Þ
P0 M
g11 ¼ þ 1:0 0 ð2:28Þ
PU M P
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2.3 Problem Formulation 17
where tW is the web thickness, HS and L are the overall depth and the span of the
cellular beam, and S is the distance between centers of holes. MU is the maximum
moment under the applied loads, MP is the plastic moment capacity of the cellular
beam, VMAXSAP is the maximum shear at support, VOMAX is the maximum shear at
the opening, VHMAX is the maximum horizontal shear, and MAAMAX is the
maximum moment at A–A section shown in Fig. 2.3. MWMAX is the maximum
allowable web post moment, VTEE represents the vertical shear on top of the hole,
P0 and M are the internal forces on the web section, and YMAX denotes the
maximum deflection of the cellular beam (Erdal et al. [10], AISC-LRFD [14]).
In design of castellated beams with hexagonal openings, the design vector includes
four design variables: the selection of a rolled beam, the selection of a cutting depth,
the spacing between the center of holes and total number of holes in the beam, and
the cutting angle as shown in Fig. 2.1. Hence the optimum design problem is
formulated by the following expression:
Find an integer design vector fXg ¼ fx1 ; x2 ; x3 ; x4 gT where x1 is the sequence
number of the rolled steel profile in the standard sections’ list, x2 is the sequence
number for the cutting depth which contains various values, x3 is the total number
of holes for the castellated beam, and x4 is the cutting angle. Thus, the design
problem turns out to be as follows:
Minimize Eq. (2.10), Eq. (2.11)
Subjected to
3
g1 ¼ d ðH S 2tf Þ 0 ð2:30Þ
8
g2 ¼ ðHS 2tf Þ 10 ðd T tf Þ 0 ð2:31Þ
2
g3 ¼ d cot θ e 0 ð2:32Þ
3
g4 ¼ e 2d cot θ 0 ð2:33Þ
g5 ¼ 2d cot θ þ e 2d 0 ð2:34Þ
∘
g6 ¼ 45 θ 0 ð2:35Þ
∘
g7 ¼ θ 64 0 ð2:36Þ
g8 ¼ M U M P 0 ð2:37Þ
g9 ¼ V MAXSUP PV 0 ð2:38Þ
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18 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
where tf is the flange thickness, dT is the depth of the tee section, MP is the plastic
moment capacity of the castellated beam, MAAMAX is the maximum moment at
A–A section shown in Fig. 2.3, MWMAX is the maximum allowable web post
moment, VTEE is the vertical shear on the tee, MTP is the moment capacity of the
tee section, and YMAX denotes the maximum deflection of the castellated beam with
hexagonal opening (Soltani et al. [3]).
In this section, the new metaheuristic algorithm developed by Kaveh and Zolghadr
[15, 16] is briefly introduced. The TWO is a population-based search method,
where each agent is considered as a team engaged in a series of tug of war
competitions. The weight of the teams is determined based on the quality of the
corresponding solutions, and the amount of pulling force that a team can exert on
the rope is assumed to be proportional to its weight. Naturally, the opposing team
will have to maintain at least the same amount of force in order to sustain its grip of
the rope. The lighter team accelerates toward the heavier team, and this forms the
convergence operator of the TWO. The algorithm improves the quality of the
solutions iteratively by maintaining a proper exploration/exploitation balance
using the described convergence operator. A summary of this method is provided
in the following steps.
Step 1: Initialization
The initial positions of teams are determined randomly in the search space:
where x0ij is the initial value of the jth variable of the ith candidate solution; xj,max
and xj,min are the maximum and minimum permissible values for the jth variable,
respectively; rand is a random number from a uniform distribution in the interval
[0, 1]; and n is the number of optimization variables.
Step 2: Evaluation of Candidate Designs and Weight Assignment The objec-
tive function values for the candidate solutions are evaluated and sorted. The best
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2.4 Optimization Algorithm 19
solution so far and its objective function value are saved. Each solution is consid-
ered as a team with the following weight:
fitðiÞ fitworst
W i ¼ 0:9 þ 0:1 i ¼ 1, 2, . . . , N ð2:46Þ
fitbest fitworst
where fit(i) is the fitness value for the ith particle. The fitness value can be
considered as the penalized objective function value for constrained problems;
fitbest and fitworst are the fitness values for the best and worst candidate solutions
of the current iteration. According to Eq. (2.46) the weights of the teams range
between 0.1 and 1.
Step 3: Competition and Displacement In TWO each team competes against all
the others one at a time to move to its new position. The pulling force exerted by a
team is assumed to be equal to its static friction force (Wμs). Hence the pulling force
between the teams i and j (Fp,ij) can be determined as max{Wiμs, Wjμs}. Such a
definition keeps the position of the heavier team unaltered.
The resultant force affecting team i due to its interaction with heavier team j in
the kth iteration can then be calculated as follows:
where Fkp;ij is the pulling force between teams i and j in the kth iteration and μk is
coefficient of kinematic friction.
!
Frk, ij
aijk ¼ gijk ð2:48Þ
W ik μk
in which akij is the acceleration of team i toward team j in the kth iteration and gkij is
the gravitational acceleration constant defined as
where Xkj and Xki are the position vectors for candidate solutions j and i in the kth
iteration. Finally, the displacement of team i after competing with team j can be
derived as
1
ΔXijk ¼ aijk Δt2 þ αk ðXmax Xmin Þ∘ð0:5 þ rand ð1; nÞÞ ð2:50Þ
2
The second term of Eq. (2.50) induces randomness into the algorithm. This term
can be interpreted as the random portion of the search space traveled by team
i before it stops after the applied force is removed. Here, α is a constant chosen from
the interval [0,1]; Xmax and Xmin are the vectors containing the upper and lower
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20 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
X
N
ΔXik ¼ ΔXijk ð2:51Þ
j¼1
The new position of team i at the end of the kth iteration is then calculated as
Step 4: Handling of Side Constraints
It is possible for the candidate solutions to leave the search space, and it is important
to deal with such solutions properly. This is especially the case for the solutions
corresponding to lighter teams for which the values of ΔX are usually bigger.
Different strategies might be used in order to solve this problem. In this study, it
is assumed that such candidate solution can be simply brought back to their
previous permissible position (Flyback strategy) or they can be regenerated
randomly.
Step 5: Termination
Steps 2 through 5 are repeated until a termination criterion is satisfied.
Flowchart of the TWO algorithm is shown in Fig. 2.4.
The pseudo-code for design of castellated beam using the tug of war optimiza-
tion algorithm is shown in Fig. 2.5. It should be noted that each team is considered a
beam.
In this section, numerical results are presented to demonstrate the efficiency of the
new metaheuristic method (TWO) for design of castellated beams. For this purpose,
three beams are selected from literature that have previously been optimized by
other algorithms. Among the steel sections’ list of British Standards, 64 universal
beam (UB) sections starting from 254 102 28 UB to 914 419 388 UB are
chosen to constitute the discrete set of steel sections from which the design
algorithm selects the cross-sectional properties for the castellated beams. In the
design pool of holes diameters, 421 values are arranged which vary between
180 and 600 mm with an increment of 1 mm. Also, for cutting depth of hexagonal
opening, 351 values are considered which vary between 50 and 400 mm with an
increment of 1 mm and cutting angle changes from 45 to 64. Another discrete set is
arranged for the number of holes. Likewise, in all the design problems, the modulus
of elasticity is equal to 205 GPa and Grade 50 is selected for the steel of the beam
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2.5 Test Problems and Optimization Results 21
The termination
Stop Yes No
Conditions satisfied?
which has the design strength of 355 MPa. The coefficients P1, P2, and P3 in the
objective function are considered as 0.85, 0.30, and 1.00, respectively (Kaveh and
Shokohi [17–20]). A maximum number of iterations of 200 are used as the
termination criterion in all the examples, and α is taken as 0.1 for all design
problems. Also, all design problems have been solved in two cases, with and
without filled holes.
A simply supported beam with a span of 4 m is considered as the first test problem,
shown in Fig. 2.6. The beam is subjected to 5 kN/m dead load including its own
weight. A concentrated live load of 50 kN also acts at mid-span of the beam, and the
allowable displacement of the beam is limited to 12 mm. For this problem the
number of agents (teams) is taken as 20.
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22 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
procedure Design of a Castellated Beam using the Tug of War Optimization algorithm
begin
Initialize parameters; Such as NOA, NOV, ROV, …% NOA=Number of Agent(Team),
NOV=Number of Variable, ROV=Range of Variable.
Generate a population of NOA random candidate solutions (Beams);
while (not termination condition) do
Analyze beams and evaluate the objective function values for them.
Define the weights of the teams (Beams) Wi based on fit(Xi)
Sort the solutions and save the best one so far.
for each team i
for each team j
if (Wi < Wj)
Move team i towards team j using Eq. (2.50);
end if
end for
Calculate the total displacement of team i using Eq. (2.51);
Determine the final position of team i using X ik +1 = X ik + ΔX ik
Use the side constraint handling technique to regenerate violating variables
Determine the new objective function for each team according to the new
positions and save the best result.
end for
end while
end
Fig. 2.5 The pseudo-code for design of castellated beam using the TWO algorithm
Castellated beams with hexagonal and circular openings are separately designed
with TWO. These beams are designed for two cases. In case 1, it is assumed that the
end of the beams is not filled. Thus the objective function for this case is obtained
from Eq. (2.10). In the second case, it is assumed that the holes in the end of the
beam are filled with steel plate, and Eq. (2.11) is utilized for the objective function.
The optimum results obtained by TWO are given in Table 2.1. It is apparent from
the same table that the optimum cost for castellated beam with hexagonal hole is
equal to 89.73$ which is obtained by TWO. Also, according to the results, the tug of
war optimization algorithm has good performance in design of cellular beam. These
results indicate that the castellated beam with hexagonal opening has less cost in
comparison to the cellular beam. The same conclusion can be drawn for the filled
opening configuration from the results listed in Table 2.1.
Figure 2.7 shows the convergence curves of the TWO algorithm for design of
castellated beams with different shapes for the openings.
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2.5 Test Problems and Optimization Results 23
In the second problem, the tug of war optimization algorithm is used to design a
simply supported castellated beam with a span of 8 m. Similar to the first example,
this beam is also designed for two different cases. The beam carries a uniform dead
load 0.40 kN/m, which includes its own weight. In addition, it is subjected to two
concentrated loads as shown in Fig. 2.8. The allowable displacement of the beam is
limited to 23 mm, and the number of agents is taken as 20.
This beam is designed by TWO, and the results are compared to those of the
other optimization algorithms as shown in Table 2.2. In design of the beam with
hexagonal hole, the corresponding cost obtained by the TWO is equal to 718.2$
which is the lowest value among all the methods. Therefore, the performance of the
tug of war optimization is better than other approaches (Kaveh and Shokohi [17–20])
for this design example. According to the obtained results, the designed beam with
hexagonal opening has less cost in comparison with the cellular beam, and it is a
better option in this case. In design of end-filled case, it is obvious that the presented
method has the same performance. Furthermore, the maximum value of the strength
ratio is equal to 0.99 for both hexagonal and circular beams, and it is shown that
these constraints are dominant in the design process.
Figure 2.9 shows the convergence history for optimum design of hexagonal
beam which is obtained by different metaheuristic algorithms.
The beam with 9 m span is considered as the last example of this study in order to
compare the minimum cost of the castellated beams. The beam carries a uniform
load of 40 kN/m including its own weight and two concentrated loads of 50 kN as
shown in Fig. 2.10. The allowable displacement of the beam is limited to 25 mm,
and the number of agent is taken as 20.
Table 2.3 compares the results obtained by the TWO with those of the other
algorithms. In the optimum design of castellated beam with hexagonal hole, TWO
algorithm selects 684 254 125 UB profile, 16 holes, and 231 mm for the cutting
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24
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2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
2.5 Test Problems and Optimization Results 25
300
TWO-H(case 1)
TWO-C(case 1)
250
TWO-H(case 2)
TWO-C(case 2)
200
Cost$
150
100
50
0 20 40 60 80 100 120 140 160 180 200
Iteration
Fig. 2.7 Convergence curves recorded in the 4 m span beam problem for the TWO best
optimization runs [1]
depth and 57 for the cutting angle. The minimum cost of the design beam is equal
to 991.04$. Also, in the optimum design of cellular beam, the TWO algorithm
selects 610 229 125 UB profile, 14 holes of diameter 490 mm. It can be
observed from Table 2.3 that the optimal design has the minimum cost of 990.33
$ for beam with hexagonal holes which is obtained by the CBO-PSO algorithm;
however, the TWO results in better design for cellular beam. In the design of beam
with filled holes, the obtained results using the tug of war optimization algorithm
are slightly different from each other. This shows that in the case of holes filled with
steel plates, where the beam span is large, using cellular beams can be a good design
strategy. Similar to the previous example, the strength criteria are dominant in the
design of this beam, and it is related to the Vierendeel mechanism. The maximum
ratio of these criteria is equal to 0.99 for both hexagonal and cellular cases.
The optimum shapes of the hexagonal and circular openings with unfilled holes
are separately shown in Fig. 2.11. Also, the convergence histories of metaheuristics
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26
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2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
2.6 Concluding Remarks 27
6000
TWO-H(case 1)
5000 ECSS-H(case 1)
CBO-H(case 1)
4000 CBO-PSO-H(case 1)
Cost$
3000
2000
1000
0
0 20 40 60 80 100 120 140 160 180 200
Iteration
Fig. 2.9 Comparison of best run convergence curves recorded in the 8 m span beam problem
(unfilled hexagonal holes) for different metaheuristic algorithms [1]
are shown in Fig. 2.12 for design of cellular beam with filled openings. It is apparent
from the figure that TWO has good convergence rate in design of this problem and
finds better solution for cellular beam.
In this chapter, the newly developed metaheuristic algorithm so-called tug of war
optimization is utilized for optimum design of castellated beams. Three benchmark
problems are solved in order to assess the robustness and efficiency of the TWO.
These beams are designed for two cases, with filled openings and unfilled openings,
where the hexagonal and circular holes are considered as the types of the web-
openings. Comparing the results obtained by TWO with those of other optimization
methods demonstrates that TWO has a better performance in the ability of finding
the optimum solution. Also, the convergence rate of this algorithm to the optimal
solution is quite good for most of problems, and it requires a less number of
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28
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2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
2.6 Concluding Remarks 29
Fig. 2.11 Optimum profiles of the castellated beams with unfilled cellular and hexagonal open-
ings for beam with 9 m span
2000
TWO-C(case 2)
ECSS-C(case 2)
1800
CBO-C(case 2)
CBO-PSO-C(case 2)
1600
Cost$
1400
1200
1000
0 20 40 60 80 100 120 140 160 180 200
Iteration
Fig. 2.12 Comparison of best run convergence curves recorded in the 9 m span beam problem
( filled circular holes) for different metaheuristic algorithms [1]
analyses to find better solution making TWO computationally more efficient. From
the results obtained in this chapter, it can be concluded that the use of the beam with
hexagonal openings leads to the use of less steel material and it is a better choice
than cellular beam in unfilled cases. For design of castellated beam with large
spans, especially in filled cases, it is observed that the cellular beam has a better
performance and it can be used as an alternative to castellated beam with hexagonal
opening.
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30 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm
References
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Chapter 3
Optimum Design of Multi-span Composite
Box Girder Bridges Using Cuckoo Search
Algorithm
3.1 Introduction
Composite steel–concrete box girders are frequently used in bridge construction for
their economic and structural advantages. An integrated metaheuristic based opti-
mization procedure is proposed for discrete size optimization of straight multi-span
steel-box girders with the objective of minimizing the self-weight of the girder. The
selected metaheuristic algorithm is the cuckoo search (CS) algorithm. The optimum
design of a box girder is characterized by geometry, serviceability, and ultimate
limit states specified by the American Association of State Highway and Transpor-
tation Officials (AASHTO). Size optimization of a practical design example inves-
tigates the efficiency of this optimization approach and leads to around 15 % of
saving in material (Kaveh et al. [1]).
For every product designed to satisfy human needs, the creator tries to achieve
the best solution for the task in hand (safety and serviceability) and therefore
performs optimization. This chapter is concerned with discrete size optimization
of straight multi-span steel-box girders with the objective of minimizing the self-
weight of girder. Composite steel-box girders in the form of built-up steel-box
sections and concrete deck slabs have become very frequent due to some positive
structural features such as high torsional and wrapping rigidity, aesthetical appeal
with regard to relatively large span-depth ratio, and economical advantages in
fabrication and maintenance (Chen and Yen [2]). Developments in computer
hardware and software, advances in computer-based analysis and design tools,
and advances in numerical optimization methods make it possible to formulate
design of complicated discrete engineering problems as optimization problems
and solve them by one of the optimization methods (Rana et al. [3]). Further
developments on box girders can be found in the works of Ding et al. [4] and Ko
et al. [5].
Many optimization methods have been developed during the last decades
pioneered by the traditional mathematical-based methods which use the gradient
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3.2 Design Optimization Problem 33
tw Dw
100° 100°
tb
After the topology and support conditions are established, the girder is divided into
some segments along the girder length. The process of division is based on
fabrication requirements. The main design effort involves sizing the individual
girder sections for the predetermined segments with the objective of minimizing
the self-weight of the girder. A typical section for composite steel–concrete box
girder is shown in Fig. 3.1. As it is depicted, the design variables in each section are
slab thickness (tc), top flange width (bf), top flange thickness (tf), web depth (Dw),
web thickness (tw), and bottom flange thickness (tb). The center to center distance of
the top flanges and the inclination angle of web from the vertical direction are fixed
to 160 cm and 100 , respectively, for the entire girder because of fabrication
conditions. As a result, the width of bottom flange is a function of other variables.
The design procedure based on the AASHTO Division I [18] provisions can be
outlined as follows:
3.2.1 Loading
Maximum compressive and tensile stresses in girders that are not provided with
temporary supports during the placing of the permanent dead load are the sum of the
stresses produced by the dead loads acting on the steel girders alone and the stresses
produced by the superimposed loads acting on the composite girder. Therefore, two
different dead loads should be considered. In the first case, the dead load is exerted
on the non-composite section (L1). This load involves self-weight of the steel girder
and weight of the concrete deck. The second case is applied on the composite
section which includes the pavement, curb, pedestrian, and guard fence loads (L2).
The highway live loads on the roadways of bridges or incidental structures shall
consist of standard trucks or lane loads that are equivalent to truck trains. AASHTO
HS loading is applied in this study. The live load for each box girder (L3) shall be
determined by applying to the girder the fraction WL of a wheel load determined by
the following equation:
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34 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
in which Nw is the number of lanes. Dynamic effects of live load should be taken
into account as an impact coefficient based on Article 3.8.2 from the
AASHTO [18].
The flanges of section, both top and bottom, should be designed for flexural
resistance as follows:
σ top
g5 : 10
σ all ðtopÞ
σ top ð3:3Þ
g6 : 10
σ all ðbotÞ
The flexural stresses of top and bottom flanges, σ(top) and σ(bot), are calculated
under three loading conditions: the section without considering concrete slab under
L1, the composite section under L2 with creep and shrinkage effects, and the
composite section under live loads without long-term effects. Creep and shrinkage
effects are taken into account by dividing concrete elastic modulus by 3 based on
10.38.1.4 (AASHTO [18]). The allowable stress of top flange, σ all(top), and tensile
allowable stress of bottom flange, σ all(bot), are equal to 0.55 Fy. The bottom flange
allowable compressive stress is supplied on the 10. 39. 4. 3.
Concrete compressive stress under L2 and L3 loads should satisfy the following
constraint:
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3.3 Parallel Metaheuristic Based Optimization Technique 35
σ concrete
g7 : 0 10 ð3:4Þ
0:4f c
0
in which fc is concrete cylindrical compressive strength.
Shear stresses in the web should be bounded by allowable shear stress as follows:
f v ¼ 2Dw tVw cos θ
g8 : 10 ð3:5Þ
Fv
where V is the shear under dead and live loads (all three load conditions) and θ is the
inclination angle of the web, fv is the shear stress, and Fv is the allowable shear
stress which is obtained by 10.39.3.1.
Complying with Sect. 10.6, the composite girder deflections under live load plus the
live load impact (ΔL+I) for each span shall not exceed 1/800 span length (S) which
can be presented as follows:
800 ΔLþI
g9 : 10 ð3:6Þ
S
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36 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
optimum design of frames (Yang and Deb [11]). The pseudo-code of the optimum
design algorithm is as follows (Kaveh and Bakhshpoori [15]):
The CS parameters are set in the first step. These parameters consist of the number
of nests (n), the step size parameter (α), the discovering probability (pa), and the
maximum number of frame analyses as the stopping criterion.
The initial locations of the nests are determined by the set of values randomly
assigned to each decision variable as
ð0Þ
nesti, j ¼ ROUND xj, min þ rand: xj, max xj, min ð3:7Þ
where nesti,j(0) determines the initial value of the jth variable for the ith nest, xj,min
and xj,max are the minimum and the maximum allowable values for the jth variable,
and rand is a random number in the interval [0, 1]. The rounding function is utilized
due to the discrete nature of the problem.
In this step, all the nests except for the best one are replaced based on quality by
new cuckoo eggs produced with Lévy flights from their positions as
ðtþ1Þ ðtÞ ðtÞ ðtÞ
nesti ¼ nesti þ α : S: nesti nestbest : r ð3:8Þ
where nestit is the ith nest current position, α is the step size parameter, r is a random
number from a standard normal distribution and nestbest is the position of the best
nest so far, and S is a random walk based on the Lévy flights. The Lévy flight
essentially provides a random walk while the random step length is drawn from a
Lévy distribution. In fact, Lévy flights have been observed among foraging patterns
of albatrosses, fruit flies, and spider monkeys. One of the most efficient and yet
straightforward ways of applying Lévy flights is to use the so-called Mantegna
algorithm. In Mantegna algorithm, the step length S can be calculated by
u
S¼ ð3:9Þ
jvj1=β
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3.3 Parallel Metaheuristic Based Optimization Technique 37
where β is a parameter between [1, 2] interval and considered to be 1.5; u and v are
drawn from normal distribution as
u N 0, σ 2u , v N 0; σ 2v ð3:10Þ
( )1=β
Γ ð1 þ βÞ sin ðπβ=2Þ
σu ¼ , σv ¼ 1 ð3:11Þ
Γ ½ð1 þ βÞ=2 β 2ðβ1Þ=2
The alien egg discovery is performed for each component of each solution in terms
of probability matrix such as
1 if rand < pa
Pij ¼ ð3:12Þ
0 if rand pa
where randperm1 and randperm2 are random permutation functions used for
different row permutations applied on nest matrix and P is the probability matrix.
The generating new cuckoos and discovering alien eggs steps are alternatively
performed until a termination criterion is satisfied. The maximum number of
analyses is considered as termination criterion of the algorithm.
A visit to the neighborhood PC retail store provides ample proof that we are in the
multi-core era. This created demand for software infrastructure to utilize mechanisms
such as parallel computing to exploit such architectures. In this respect, the
MathWorks introduced Parallel Computing Toolbox software and MATLAB® Dis-
tributed Computing Server (Luszczek [9]). Regarding that our individual designs
proposed by population-based metaheuristic algorithms are evaluated independently,
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38 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
electing one of MATLAB’s most basic programming paradigms, the parallel for loops
(Luszczek [9]), makes it easy for user to handle such optimization problem.
Since the parallel computing technique enables us to perform several actions at
the same time, it is needed to adjust the analysis and design assumptions for a prime
model of box girder in the SAP2000 environment. Once the optimization algorithm
invokes the model, a set of sections are assigned to the predefined segments. A
certain feasible number of proposed solutions get invoked for analysis, and evalu-
ating the penalized fitness value following the PARFOR conditional command the
next set of agents is generated. The iteration continues until a stopping criterion is
attained.
where {X} is the set of design variables and its components are sized from the discrete
sets presented in Table 3.2 and W({X}) is the self-weight of girder obtained by
SAP2000. Optimum design of composite steel-box girders is one of those issues for
which the conventional objective function is not applicable. Considering concrete slab,
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3.4 Design Example 39
(a)
S1 S2 S3 S4 S5 S6 S7 S8
3m
9m 6m 6m 4m 6m 6m 6m 6m 12m
(b) 11.80
0.35 10.40 0.70 0.35
0.35 0.70 0.35
Fig. 3.2 The practical design example. (a) Longitudinal view and (b) transverse view
shear connectors, and reinforcement cost seems to be necessary. Cost of the shear
connectors is negligible in comparison to the overall cost. Higher strength shear
connectors are considered to satisfy the complete composite action. According to
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40 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
Articles 3.24.10.2 and 3.24.3.1 provided by AASHTO [18] for designing the longitudi-
nal and transverse reinforcement, the reinforcement depends only on the slab thickness
and the distance of the girders. Thus reinforcement is not considered as a design
variable. Considering the concrete slab thickness as a design variable, the proposed
objective function is not representative and needs to be modified. Instead of the total
weight (concrete slab weight and steel section weight altogether), the sum of the total
cost of the concrete material and the total cost of the steel section material should be
used. Modification can be made using unit cost coefficients for each item. The choice of
the unit cost parameters can influence the properties of the most cost-efficient design
(Fragiadakis and Lagaros [19]). In addition slab thickness as a design variable has a
profound effect on the model stiffness matrix and dead load. Considering tc as a design
variable simultaneously with design variables representing the steel section can lead the
algorithm to unfeasible designs. In these regards, the CS is applied to find the optimum
design considering the slab thickness as a constant value from a certain practical interval
[0.2, 0.35] with 0.05 m increment to achieve the optimum thickness. The lower bound is
considered according to the provisions of AASHTO [18] (Table 3.8.9.2).
The design should be carried out in such a way that the girder satisfies the
strength, displacements, and geometric requirements presented in the second sec-
tion. In order to handle the constraints, a penalty approach is utilized. In this
method, the aim of the optimization is redefined by introducing the cost function as
where N is the constraint violation function. For generating the total penalty, each
segment is divided into five equal parts, and all the constraints, g1 to g8, are checked
for each part. In this way the constraint violation function can be obtained as
follows:
X
8
N¼ vi , vi ¼ max μj , j ¼ 1, 2, ::, 5
i¼1
ð3:16Þ
X
9
μj ¼ max½gk ; 0
k¼1
in which νi is the penalty of each segment and μj is the penalty value for jth part of
ith segment. ε1 and ε2 are penalty function exponents which are selected consider-
ing the exploration and the exploitation rate of the search space. Here, ε1 is set to
unity; ε2 is selected in a way that, in the first steps of the search process, it is equal to
1 and ultimately increased to 3.
In modeling, analysis, and design procedures, the fundamental assumptions are
made to idealize the results as follows: Material property for all sections is
considered as A36 steel material with weight per unit volume of ρ ¼ 7849 kg/m3
(0.2836 lb/in3), modulus of elasticity of E ¼ 199,948 MPa (29,000 ksi) and a yield
stress of fy ¼ 248.2 MPa (36 ksi), and concrete material with the strength of
0
fc ¼ 24 MPa (ksi) and ρ ¼ 2500 t/m3 (lb/in3); the spacing of transverse stiffeners
is assumed 2 m and the bottom flange is longitudinally stiffened. As it was
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3.4 Design Example 41
46
42
40
38 CS
PSO
36
HS
34
32
20 25 30 35
Concrete slab thickness (cm)
mentioned, the girder carries three types of loads (ton/m) as follows: L1 ¼ slab
weight + self-weight of girder, L2 ¼ 1.22, and L3 ¼ 1.326 (HS loading on a girder).
In order to verify the efficiency of the CS, two other algorithms are used to
determine the solution of the considered discrete optimization problem, which are
harmony search method (HS) (Kochenberger and Glover [16]) and PSO (Kennedy
et al. [17]) algorithm. These algorithms have been frequently used in multicriteria
and constrained optimization, typically associated with practical engineering prob-
lems. For example, Erdal et al. [20] have utilized these algorithms for optimum
design of cellular beams. The author and colleagues have used these algorithms for
discrete optimum design problem similar to the work by Erdal et al. [20]. Additional
details can be found in Erdal et al. [20]. Here the PSO, HS, and CS algorithms are
used for obtaining the optimum slab thickness and two adjacent depths. Consider-
ing the effect of the initial solution on the final results and the stochastic nature of
the metaheuristic algorithms, each algorithm is independently solved for five times
with random initial designs. Then the best run is chosen for performance evaluation
of each technique. The maximum number of box girder evaluations are considered
as 7000 for the termination criteria. The parameters of the CS algorithm are
considered as n ¼ 7, α ¼ 0.1, and pa ¼ 0.3. The parameters of the PSO algorithm
are tuned as NPT ¼ 50, C1 ¼ C2 ¼ 2, ω ¼ 1.2, and Vmax ¼ Δt ¼ 1.3, and the param-
eters of the HS algorithm are tuned as hms ¼ 70, hmcr ¼ 0.8, and par ¼ 0.2.
3.4.2 Discussions
Figure 3.3 shows the obtained optimum weight for various concrete slab thick-
nesses by the algorithms. All three algorithms result in the optimum thickness of
concrete slab as 0.2 m. It can be concluded that in this test problem, considering the
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42 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
800 200
CS
150 PSO
Penalized weight (tons)
600 HS
100
50
400
0 1000 2000 3000 4000 5000 6000 7000
200
0
0 1000 2000 3000 4000 5000 6000 7000
Number of girder evaluations
Fig. 3.4 Best convergence history obtained by three metaheuristic algorithms (tc ¼ 20 cm)
concrete slab thickness equal to the minimum value provided by the AASHTO [18]
provisions leads to the optimum design. The optimum feasible designs obtained by
CS, PSO, and HS algorithms weighted 32.77, 33.34, and 38.36 t, respectively. For
graphical comparison of algorithms, the convergence histories for the best result of
five independent runs in the case of tc ¼ 0.2 m are shown in Fig. 3.4. PSO and CS act
far better than the HS algorithm. PSO algorithm shows the fastest convergence rate
compared to other methods and this is because of the good global search ability of
PSO. It is obvious that PSO cannot perform efficiently in the local search stage of
the algorithm. However, PSO results in the same practical design as the CS but
needs higher number of girder evaluations (6450). Continuous step like movements
of the CS algorithm demonstrates its ability in balancing the global and local search
in this optimization test problem. The optimum design obtained by cuckoo search
algorithm is weighted 32.77 t which is approximately 15 % lighter than the con-
ventional design. Related cross-sectional properties and mass per length of sections
for each segment are summarized in Table 3.3. The cross-sectional properties based
on the conventional design, considering the concrete slab thickness equal to 0.2 cm,
are also presented in this table.
Geometry constraint values of sections for each segment are listed in Table 3.4.
As it can be seen, the first constraint (g1) with the aim of controlling the top flange
thickness to the web thickness is the most active limitation. The last row exhibits
optimum design controlling priority with respect to the geometry constraints. The
serviceability and strength performance of the resulted optimum girder are illus-
trated in Fig. 3.5. Based on this figure, in spite of relatively long middle span, the
effect of deflection constraint is not notable here. Such a performance is also
observed for shear stress ratio constraint. Figure 3.5c shows the available and
allowable flexural stress ratios for the top and bottom flanges and the concrete
deck. It can be observed that the stress ratio of top and bottom flanges have more
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3.4 Design Example 43
Table 3.3 Sectional designations of the best optimum design obtained by the CS
Mass per length
Segment Section bf tf Dw tw tb (kg/m)
S1 A1 0.3 0.02 0.7 0.01 0.015 363.83 (741.55)
(0.45) (0.025) (1.5) (0.015) (0.025)
S2 A2 0.3 0.02 0.7 0.01 0.025 470.33 (741.55)
(0.45) (0.025) (1.5) (0.015) (0.025)
A3 0.3 0.02 1.8 0.01 0.025 568.05 (703.55)
(0.45) (0.03) (2) (0.01) (0.025)
A4 0.3 0.02 1.7 0.01 0.025 559.16 (588.49)
(0.45) (0.02) (1.5) (0.01) (0.02)
S3 A5 0.3 0.015 1.7 0.01 0.01 416.75 (546.14)
(0.45) (0.02) (1.5) (0.01) (0.02)
S4 A6 0.3 0.02 1.7 0.01 0.025 559.16 (546.14)
(0.45) (0.02) (1.5) (0.01) (0.02)
S5 A7 0.3 0.02 1.7 0.01 0.015 479.92 (546.14)
(0.45) (0.02) (1.5) (0.01) (0.02)
S6 A8 0.3 0.02 1.7 0.01 0.02 519.54 (546.14)
(0.45) (0.02) (1.5) (0.01) (0.02)
A9 0.3 0.02 2.0 0.01 0.02 550.28 (703.55)
(0.45) (0.03) (2) (0.01) (0.025)
A10 0.3 0.02 0.8 0.01 0.02 427.33 (741.55)
(0.45) (0.025) (1.5) (0.015) (0.025)
S7 A11 0.3 0.015 0.8 0.01 0.015 351.89 (741.55)
(0.45) (0.025) (1.5) (0.015) (0.025)
S8 A12 0.3 0.02 0.8 0.01 0.01 323.55 (546.14)
(0.45) (0.02) (1.5) (0.01) (0.02)
The values in parentheses are the at hand design using the conventional design procedure
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44 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
(a)
0.05
Maximum allowable value
0.045
0.04
0.035
0.03
Deflection
0.025
0.02
0.015
0.01
0.005
0
0 1.8 3.6 5.4 7.2 9 15 2122.6 27.4 29.8 32.2 34.6 37 39.4 41.8 49 55 57.4 59.8 62.2 64.6 67 70
0.8
Shear stress ratio
0.6
0.4
0.2
0
0 1.8 3.6 5.4 7.2 9 15 2122.6 27.4 29.8 32.2 34.6 37 39.4 41.8 49 55 57.4 59.8 62.2 64.6 67 70
0.8
Flexural stress ratio
0.6
0.4
0.2
0
0 1.8 3.6 5.4 7.2 9 15 2122.6 27.4 29.8 32.2 34.6 37 39.4 41.8 49 55 57.4 59.8 62.2 64.6 67 70
Longitudinal direction of the girder
Fig. 3.5 Performance evaluation of the best achieved optimum design via CS. (a) Deflection; (b)
shear stress ratio; (c) flexural stress ratios
effect in controlling the optimum design than the shear and concrete slab stress
ratios. Also it can be interpreted that the bottom and the top flange stress ratios are
dominant at the middle of spans and on the supports, respectively. This can be due
to the contribution of the concrete slab in carrying the loads in a composite manner
at the middle of spans.
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References 45
References
@Seismicisolation
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46 3 Optimum Design of Multi-span Composite Box Girder Bridges Using Cuckoo. . .
11. Yang XS, Deb S (2009) Engineering optimisation by cuckoo search. Int J Math Model Numer
Optim 1:330–343
12. Kaveh A, Bakhshpoori T, Ashoori M (2012) An efficient optimization procedure based on
cuckoo search algorithm for practical design of steel structures. Int J Optim Civil Eng 2
(1):1–14
13. Saka MP, Dogan E (2012) Design optimization of moment resisting steel frames using a
cuckoo search algorithm. In Topping BHV (ed) Proceedings of the eleventh international
conference on computational structures technology. Civil-Comp Press, Stirlingshire, Paper 71.
doi:10.4203/ccp.99.71
14. Saka MP, Geem ZW (2013) Mathematical and metaheuristic applications in design optimiza-
tion of steel frame structures: an extensive review. Math Prob Eng Article ID 271031:33
15. Kaveh A, Bakhshpoori T (2013) Optimum design of steel frames using Cuckoo Search
algorithm with Lévy flights. Struct Des Tall Spec Build 22(13):1023–1036
16. Kochenberger GA, Glover F (2003) Handbook of metaheuristics. Kluwer, Boston, MA
17. Kennedy J, Eberhart R, Shi Y (2001) Swarm intelligence. Morgan Kaufmann, San Francisco,
CA
18. American Association of State Highway and Transportation Officials (AASHTO) (2002)
Standard specifications for highway bridges, 17th edn. AASHTO, Washington, DC
19. Fragiadakis M, Lagaros ND (2011) An overview to structural seismic design optimisation
frameworks. Comput Struct 89:1155–1165
20. Erdal F, Dogan E, Saka MP (2011) Optimum design of cellular beams using harmony search
and particle swarm optimizers. J Constr Steel Res 67:237–247
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Chapter 4
Sizing Optimization of Skeletal Structures
Using the Enhanced Whale Optimization
Algorithm
4.1 Introduction
where {X} is the vector containing the design variables; ng is the number of design
variables; W({X}) is the weight of the structure; nm is the number of elements of the
structure; ρi, Ai, and Li denote the material density, cross-sectional area, and the
length of the ith member, respectively; ximin and ximax are the lower and upper
bounds of the design variable xi, respectively; gj({X}) denotes design constraints;
and nc is the number of constraints.
To handle the constraints, the well-known penalty approach is employed. Thus,
the objective function is redefined as follows:
X
nc h i
f ðfXgÞ ¼ ð1 þ ε1 :υÞε2 W ðfXgÞ, υ¼ max 0, gj ðfXgÞ ð4:2Þ
j¼1
where υ denotes the sum of the violations of the design constraints. The constant ε1
is set equal to 1 while ε2 starts from 15 and then linearly increases to 3.
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4.3 Optimization Algorithms 49
∗
~
X ð t þ 1Þ ¼ ~ ~
X ðtÞ A: D ð4:3Þ
∗
~ ¼ C: ~
D XðtÞ
X ðt Þ ~ ð4:4Þ
A ¼ 2:a:r a ð4:5Þ
C ¼ 2:r ð4:6Þ
∗
where ~X is the historically best position, ~X is a whale position and t indicates the
current iteration, a is linearly decreased from 2 to 0 over the course of iterations,
and r is a random number uniformly distributed in the range of [0,1]. The sign
“||” denotes the absolute value.
2. Spiral bubble-net feeding maneuver: A spiral equation is used between the
position of whale and prey to mimic the helix-shaped movement of humpback
whales as follows:
~0 þ ~∗
~
Xðt þ 1Þ ¼ ebk : cos ð2πkÞ: D X ðtÞ ð4:7Þ
∗
~0 ¼ ~
D XðtÞ
X ðtÞ ~ ð4:8Þ
where b is a constant for defining the shape of the logarithmic spiral and k is a
random number uniformly distributed in the range of [1,1].
In order to have a global optimizer, when A is >1 or A <1, the search agent is
updated according to a randomly chosen search agent instead of the best search
agent:
00
~
X ð t þ 1Þ ¼ ~ ~
Xrand A: D ð4:9Þ
00
~ ¼ C: ~
D XðtÞ
Xrand ~ ð4:10Þ
where ~Xrand is selected randomly from whales in the current iteration. For further
details, the reader may refer to Mirjalili and Lewis [2].
The WOA is simple in concept and effective to explore global solutions. In order to
improve the solution accuracy, reliability of search, and convergence speed of
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50 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
WOA, a new algorithm is introduced in this chapter, which is called the EWOA. A key
point in improving an algorithm is to preserve the simplicity of the original method.
A random number in the [0,1] range is extracted for each whale in each iteration.
If it is >0.5, Eq. (4.7) is selected; otherwise, Eq. (4.12) is chosen for updating
whale’s position.
In exploration phase of EWOA, one component of each whale is changed with
the random value in the search space with a probability like p instead of Eq. (4.9).
where iter and itermax are current iteration number and the total number of the
iterations for optimization process, respectively.
For a selected whale, an integer random number is extracted in the interval [1,
ng] to choose which design variable should be randomly changed. At this point,
another random number q is extracted in the interval [0,1] and compared with the
variable xj is changed if q < p, according to
probability thresholdp. The selected
xj ¼ xjmin þ random: xjmax xjmin , where random is a random number uniformly
distributed in the interval [0, 1].
The modified algorithm should be capable of maintaining proper balance
between the diversification and the intensification inclinations. According to this
point and the above change, Eq. (4.3) is redefined as follows:
000
∗
~
X ð t þ 1Þ ¼ ~X ðtÞ ~ ~
A∘ D ð4:12Þ
000
~ ¼~
D r∘~ XðtÞ ð4:13Þ
~
A ¼ 2:~ r~
a∘~ a ð4:14Þ
where ~r is a random vector that has each component uniformly distributed in the
a is a vector that has each component equal to a. The sign “ ”
range of [0,1] and ~
denotes an element-by-element multiplication.
Flowchart of EWOA is shown in Fig. 4.1.
In this section, four benchmark examples are provided to demonstrate the effec-
tiveness, robustness, and efficiency of the WOA and EWOA. In order to reduce
statistical errors, each test is repeated 20 times independently. In all problems,
agents are allowed to select discrete values from the permissible list of cross
sections (real numbers are rounded to the nearest integer in the each iteration).
The algorithms are coded in MATLAB, and the structures are analyzed using the
direct stiffness method by our own codes.
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4.4 Test Problems and Optimization Results 51
Is
terminating No
criterion
fulfilled?
Yes
End
Figure 4.2 shows the schematic of a spatial 72-bar truss structure. The material
density is 0.1 lb/in3 (2,767,990 kg/m3) and the modulus of elasticity is 107 psi
(6895 GPa). The elements are divided into 16 groups, because of structural sym-
metry: (1) A1–A4, (2) A5–A12, (3) A13–A16, (4) A17–A18, (5) A19– A22, (6) A23–A30,
(7) A31–A34, (8) A35–A36, (9) A37–A40, (10) A41–A48, (11) A49–A52, (12) A53–A54,
(13) A55–A58, (14) A59–A66 (15), A67– A70, and (16) A71–A72. The structure is
subject to the two independent loading conditions listed in Table 4.1. The maxi-
mum stress developed in the elements must be less than 25 ksi (172,375 MPa).
Maximum displacement of the uppermost nodes cannot exceed 0.25 in
(635 mm), for each node, in all directions. In this case, the discrete sizing
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52 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
variables can be selected from a list of 64 discrete sections from 0.111 to 335 in2
(71,613–21,612,860 mm2) (Kaveh and Ilchi Ghazaan [3]).
This example is also used for adjusting b [a constant for defining the shape of the
logarithmic spiral in Eq. (4.7)], number of whales, and itermax (total number of
iterations). In order to adjust the value of b, a number of whales and itermax are,
respectively, set to 20 and 1000, and different amounts of b are considered as 0.5,
1, 15, and 2. The results shown in Table 4.2 demonstrate that the algorithm is not
very sensitive to the values of b; however, statistical results indicate that 0.5 is the
most efficient value. In order to adjust the number of whales, the value of itermax is
set to 1000, and various numbers of whales are selected as 10, 20, 30, and 40.
Comparison of the results is shown in Table 4.3, and it can be seen that 20 is a quite
suitable number. Different itermax are tested (500, 750, 1000, 1250, and 1500) to
adjust this variable. Table 4.4 summarizes the results and it can be concluded 1000
is the most suitable value for itermax.
Table 4.5 represents the results obtained by different optimization algorithms.
The lightest designs obtained by discrete heuristic particle swarm ant colony
optimization (DHPSACO) (Kaveh and Talatahari [4]), imperialist competitive
algorithm (ICA) (Kaveh and Talatahari [5]), and colliding bodies optimization
(CBO) (Kaveh and Ilchi Ghazaan [3]) are 393,380 lb, 39,284 lb, and 39,123 lb,
respectively. The best designs of improved ray optimization (IRO) (Kaveh et al.
[6]), adaptive elitist differential evolution (aeDE) (Ho-Huu et al. [7]), WOA, and
EWOA are identical (i.e., 38,933 lb). EWOA was the most robust optimizer,
achieving the lowest average weight over the independent optimization runs.
Figure 4.3 shows the convergence curves of the best and average results obtained
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4.4 Test Problems and Optimization Results 53
Table 4.1 Loading conditions for the spatial 72-bar truss problem
Condition 1 Condition 2
Node Fx kips (kN) Fy kips (kN) Fz kips (kN) Fx kips (kN) Fy kips (kN) Fz kips (kN)
17 0.0 0.0 5.0 (22.25) 5.0 (22.25) 5.0 (22.25) 5.0 (22.25)
18 0.0 0.0 5.0 (22.25) 0.0 0.0 0.0
19 0.0 0.0 5.0 (22.25) 0.0 0.0 0.0
20 0.0 0.0 5.0 (22.25) 0.0 0.0 0.0
Table 4.2 Sensitivity of EWOA to the b parameter studied for the 72-bar truss problem
Results
b Weight (lb) Average optimized weight (lb) Standard deviation on average weight (lb)
0.5 389.33 389.64 0.74
1 389.33 389.98 1.58
1.5 389.33 389.89 1.29
2 389.33 389.81 0.78
Table 4.3 Sensitivity of EWOA to the number of whales studied for the 72-bar truss problem
Results
Number of Average optimized Standard deviation on average
whales Weight (lb) weight (lb) weight (lb)
10 389.33 390.03 1.36
20 389.33 389.64 0.74
30 389.33 389.73 0.71
40 389.33 389.86 0.97
Table 4.4 Sensitivity of EWOA to the itermax parameter studied for the 72-bar truss problem
Results
Average optimized Standard deviation on average
itermax Weight (lb) weight (lb) weight (lb)
500 389.33 390.28 1.89
750 389.33 390.49 1.53
1000 389.33 389.64 0.74
1250 389.33 389.90 0.95
1500 389.33 389.93 1.33
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Table 4.5 Optimized designs found by different algorithms in the 72-bar truss problem
54
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weight (lb)
Number of struc- 5330 4500 17,925 4620 4160 6960 10,460
tural analyses
Constraint None None None None None None None
4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
tolerance (%)
4.4 Test Problems and Optimization Results 55
Fig. 4.3 Convergence curves obtained by EWOA and WOA in the 72-bar truss problem [1]
by WOA and EWOA. The best designs have been located at 6960 and 10,460
analyses for WOA and EWOA, respectively.
The spatial 582-bar tower truss shown in Fig. 4.4 is optimized for minimum volume
with the cross-sectional areas of the members being the design variables. The
582 members are divided into 32 groups, because of structural symmetry. Cross-
sectional areas of elements (sizing variables) are selected from a discrete list of
W-shaped standard steel sections based on area and radii of gyration properties.
Cross-sectional areas of elements can vary between 616 and 215 in2 (i.e., between
3974 and 138,709 cm2). A single load case is considered: lateral loads of 112 kips
(50 kN) applied in both x- and y-directions and a vertical load of 674 kips
(30 kN) applied in the z-direction at all nodes of the tower. Limitation on stress
and stability of truss elements are imposed according to the provisions of AISC [8]
as follows:
The allowable tensile stresses for tension members are calculated as
σþ
i ¼ 0:6Fy ð4:15Þ
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56 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
8" ! # " #
>
> λ 2
5 3λ λ 3
>
> 1 2 Fy = þ
i i
i
for λi < C
< 2Cc 3 8Cc 8C3c
σi ¼ ð4:16Þ
>
> 12π 2 E
>
> for λi Cc
:
23λ2i
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Table 4.6 Optimized designs found by different algorithms in the 582-bar tower problem
Cross-sectional areas (in2)
aeDE Kaveh and Ilchi
DHPSACO (Kaveh ICA (Kaveh and IRO (Kaveh CBO (Kaveh and (Ho-Huu et al. Ghazaan [1]
Element group Members and Talatahari [4]) Talatahari [5]) et al. [6]) Ilchi Ghazaan [3]) [7]) WOA EWOA
1 A1–A4 1.800 1.99 1.99 2.13 1.99 1.99 1.99
2 A5–A12 0.442 0.442 0.563 0.563 0.563 0.563 0.563
3 A13–A16 0.141 0.111 0.111 0.111 0.111 0.111 0.111
4 A17–A18 0.111 0.141 0.111 0.111 0.111 0.111 0.111
5 A19–A22 1.228 1.228 1.228 1.228 1.228 1.228 1.228
6 A23–A30 0.563 0.602 0.563 0.442 0.442 0.442 0.442
7 A31–A34 0.111 0.111 0.111 0.141 0.111 0.111 0.111
8 A35–A36 0.111 0.141 0.111 0.111 0.111 0.111 0.111
9 0.563 0.563 0.563 0.442 0.563 0.563 0.563
4.4 Test Problems and Optimization Results
A37–A40
10 A41–A48 0.563 0.563 0.442 0.563 0.563 0.563 0.563
11 A49–A52 0.111 0.111 0.111 0.111 0.111 0.111 0.111
12 A53–A54 0.250 0.111 0.111 0.111 0.111 0.111 0.111
13 A55–A58 0.196 0.196 0.196 0.196 0.196 0.196 0.196
14 A59–A66 0.563 0.563 0.563 0.563 0.563 0.563 0.563
15 A67–A70 0.442 0.307 0.391 0.391 0.391 0.391 0.391
16 A71–A72 0.563 0.602 0.563 0.563 0.563 0.563 0.563
Weight (lb) 393.380 392.84 389.33 391.23 389.33 389.33 389.33
Average opti- N/A N/A 408.17 456.69 390.913 392.52 389.64
mized weight (lb)
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Worst optimized N/A N/A N/A N/A 393.325 399.65 391.83
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weight (lb)
Number of struc- 5330 4500 17,925 4620 4160 6960 10,460
tural analyses
Constraint None None None None None None None
57
tolerance (%)
58 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
1,302,038 in3, respectively. EWOA was again the most robust optimizer, achieving
the lowest average volume over the independent optimization runs. The stress ratios
evaluated for the best design optimized by WOA and EWOA are shown in Fig. 4.5.
The maximum stress ratio and the maximum nodal displacements obtained by WOA
are 99.87 % and 31,499 in, respectively, while 99.90 % and 31,497 in are found by
EWOA for maximum stress ratio and the maximum nodal displacements. Figure 4.6
illustrates the convergence curves found by the proposed methods. The best designs
Fig. 4.5 Stress ratios evaluated at the optimized designs found by EWOA and WOA in the
582-bar tower problem
Fig. 4.6 Convergence curves obtained by EWOA and WOA in the 582-bar tower problem [1]
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4.4 Test Problems and Optimization Results 59
are achieved after 18,840 and 19,300 analyses in WOA and EWOA, respectively.
However, EWOA required only about 14,000 analyses to find better intermediate
designs than WOA and 17,100 analyses to find an intermediate design with volume
1,302,000 in3, better than the WOA optimized volume (1,302,038 in3). Furthermore,
EWOA required only 11,740 analyses to find a volume of 1,330,000 in3, better than
the design optimized by CBO (1,334,994 in3 within 17,700 analyses).
Figure 4.7 represents the schematic of a 3-bay 15-story frame. The applied loads
and the numbering of member groups are also shown in this figure. The modulus of
elasticity is 29 Msi (200 GPa) and the yield stress is 36 ksi (2482 MPa). The
effective length factors of the members are calculated as kx 0 for a sway-
permitted frame, and the out-of-plane effective length factor is specified as
ky ¼ 10. Each column is considered as non-braced along its length, and the
non-braced length for each beam member is specified as one-fifth of the span
length. Limitation on displacement and strength is imposed according to the pro-
visions of AISC [12] as follows:
(a) Maximum lateral displacement
ΔT
R0 ð4:17Þ
H
di
RI 0, i ¼ 1, 2, . . . , ns ð4:18Þ
hi
where di is the inter-story drift, hi is the story height of the ith floor, ns is the
total number of stories, and RI is the inter-story drift index (1/300).
(c) Strength constraints
8
> Pu Mu Pu
>
< 2φ P þ φ M 1 0, for < 0:2
c n b n φc P n
ð4:19Þ
>
> P 8M Pu
: u
þ
u
1 0, for 0:2
φc Pn 9φb Mn φc P n
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60 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
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4.4 Test Problems and Optimization Results 61
the nominal flexural strength, and φb denotes the flexural resistance reduction
factor (φb ¼ 0.90). The nominal tensile strength for yielding in the gross section
is calculated by
P n ¼ A g : Fy ð4:20Þ
Pn ¼ Ag : Fcr ð4:21Þ
where
8
> λ2c
< Fcr ¼ 0:658 ! Fy ,
> for λc 1:5
0:877 ð4:22Þ
>
>
: Fcr ¼ λ2c
Fy , for λc > 1:5
rffiffiffiffiffi
kl Fy
λc ¼ ð4:23Þ
rπ E
where GA and GB are stiffness ratios of columns and girders at two end joints,
A and B, of the column section being considered, respectively.
Also, the sway of the top story is limited to 925 in (235 cm) in this example.
The designs optimized by HPSACO (heuristic particle swarm ant colony opti-
mization) (Kaveh and Talatahari [14]), HBB–BC (Kaveh and Talatahari [10]), ICA
(Kaveh and Talatahari [5]), CSS (charged system search) (Kaveh and Talatahari
[15]), CBO (Kaveh and Ilchi Ghazaan [3]), WOA, and EWOA are compared in
Table 4.7. The EWOA algorithm obtained the lowest weight, which is 88,090 lb.
EWOA was the most robust optimizer also in this test problem, obtaining the lowest
average weight over the independent optimization runs. Stress ratios and inter-story
drifts evaluated for the best designs of WOA and EWOA are shown in Figs. 4.8 and
4.9. Figure 4.10 compares the best and average convergence histories of EWOA
and WOA. The best designs are achieved after 19,060 and 19,940 analyses in WOA
and EWOA, respectively.
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62
Table 4.7 Optimized designs found by different algorithms in the 3-bay 15-story frame problem
Optimal W-shaped sections
Kaveh and Ilchi Ghazaan
HPSACO (Kaveh HBB–BC (Kaveh ICA (Kaveh and CSS (Kaveh and CBO (Kaveh and [1]
Element group and Talatahari [14]) and Talatahari [10]) Talatahari [5]) Talatahari [15]) Ilchi Ghazaan [3]) WOA EWOA
1 W21 111 W24 117 W24 117 W21 147 W24 104 W14 90 W14 99
2 W18 158 W21 132 W21 147 W18 143 W40 167 W30 173 W27 161
3 W10 88 W12 95 W27 84 W12 87 W27 84 W12 79 W27 84
4 W30 116 W18 119 W27 114 W30 108 W27 114 W27 114 W24 104
5 W21 83 W21 93 W14 74 W18 76 W21 68 W14 68 W21 68
6 W24 103 W18 97 W18 86 W24 103 W30 90 W30 90 W18 86
7 W21 55 W18 76 W12 96 W21 68 W8 48 W21 48 W21 48
8 W27 114 W18 65 W24 68 W14 61 W21 68 W14 68 W14 68
9 W10 33 W18 60 W10 39 W18 35 W14 34 W8 24 W8 31
10 W18 46 W10 39 W12 40 W10 33 W8 35 W14 48 W10 45
11 W21 44 W21 48 W21 44 W21 44 W21 50 W21 44 W21 44
Weight (lb) 95,850 97,689 93,846 92,723 93,795 88,651 88,090
Average opti- N/A N/A N/A N/A 98,738 92,903 90,784
mized weight
(lb)
Worst optimized N/A N/A N/A N/A N/A 99,806 94,931
weight (lb)
Number of 6800 9900 6000 5000 9520 19,060 19,940
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structural
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analyses
Constraint toler- None None None None None None None
ance (%)
4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
4.4 Test Problems and Optimization Results 63
Fig. 4.8 Stress ratios evaluated at the optimized designs found by EWOA and WOA in the 3-bay
15-story frame problem
Fig. 4.9 Inter-story drifts evaluated at the optimized designs found by EWOA and WOA in the
3-bay 15-story frame problem [1]
Figure 4.11 shows the schematic of a 3-bay 24-story frame. Frame members are
collected in 20 groups (16 column groups and 4 beam groups). Each of the four
beam element groups is chosen from all 267 W shapes, while the 16 column
element groups are limited to W14 sections. The material has a modulus of
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64 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
Fig. 4.10 Convergence curves obtained by EWOA and WOA in the 3-bay 15-story frame
problem [1]
elasticity equal to E ¼ 29,732 Msi (205 GPa) and a yield stress of fy ¼ 334 ksi
(2303 MPa). The effective length factors of the members are calculated as kx 0 for
a sway-permitted frame, and the out-of-plane effective length factor is specified as
ky ¼ 10. All columns and beams are considered as non-braced along their lengths.
Similar to the previous example, the frame is designed following the AISC-LRFD
specifications and uses an inter-story drift displacement constraint (AISC [12]).
The optimized designs found by the different algorithms are compared in
Table 4.8. The lightest design (i.e., 203,490 lb) is again obtained by EWOA. The
best weights found by ACO (ant colony optimization) (Camp et al. [16]), HS
(harmony search) (Degertekin [17]), ICA (Kaveh and Talatahari [5]), CSS
(Kaveh and Talatahari [15]), CBO (Kaveh and Ilchi Ghazaan [3]), and WOA are,
220,465 lb, 214,860 lb, 212,640 lb, 212,364 lb, 215,874 lb, and 206,520 lb,
respectively. The average optimized weight achieved by EWOA is better than
those obtained by the other metaheuristic algorithms considered in this study.
Figure 4.12 compares the convergence curves obtained by EWOA and WOA,
which found the optimum weight after 18,820 and 19,640 structural analyses,
respectively. It should be noted that EWOA required only 10,500 analyses to find
an intermediate design weighing 210,000 lb, better than the designs optimized by
ICA and CSS (212,640 and 212,364 lb, respectively), and only 13,500 analyses to
find an intermediate design weighing 206,000 lighter than the WOA optimized
design (206,520 lb).
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4.4 Test Problems and Optimization Results 65
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Table 4.8 Optimized designs found by different algorithms in the 3-bay 24-story frame problem
66
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20 W14 26 W14 22 W14 22 W14 22 W14 22 W14 22 W14 22
Weight (lb) 220,465 214,860 212,640 212,364 215,874 206,520 203,490
Average optimized 229,555 222,620 N/A 215,226 225,071 216,475 208,648
weight (lb)
4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
Worst optimized N/A N/A N/A N/A N/A 243,143 226,019
weight (lb)
Number of struc- 15,500 13,924 7500 5500 8280 19,640 18,820
tural analyses
Constraint toler- None None None None None None None
ance (%)
4.4 Test Problems and Optimization Results
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67
68 4 Sizing Optimization of Skeletal Structures Using the Enhanced Whale. . .
Fig 4.12 Convergence curves obtained by EWOA and WOA in the 3-bay 24-story frame
problem [1]
References
1. Kaveh A, Ilchi Ghazaan M (2016) Enhanced whale optimization algorithm for sizing optimi-
zation of skeletal structures. Mech Based Des Struct Mach Int J (Published online: 21 July
2016)
2. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67
3. Kaveh A, Ilchi Ghazaan M (2015) A comparative study of CBO and ECBO for optimal design
of skeletal structures. Comput Struct 153:137–147
4. Kaveh A, Talatahari S (2009) A particle swarm ant colony optimization for truss structures
with discrete variables. J Constr Steel Res 65:1558–1568
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References 69
5. Kaveh A, Talatahari S (2010) Optimum design of skeletal structure using imperialist compet-
itive algorithm. Comput Struct 88:1220–1229
6. Kaveh A, Ilchi Ghazaan M, Bakhshpoori T (2013) An improved ray optimization algorithm for
design of truss structures. Periodica Polytech 57:1–15
7. Ho-Huu V, Nguyen-Thoi T, Vo-Duy T, Nguyen-Trang T (2016) An adaptive elitist differential
evolution for optimization of truss structures with discrete design variables. Comput Struct
165:59–75
8. American Institute of Steel Construction (AISC) (1989) Manual of steel construction: allow-
able stress design. American Institute of Steel Construction, Chicago, IL
9. Hasançebi O, Çarbas S, Dogan E, Erdal F, Saka MP (2009) Performance evaluation of
metaheuristic search techniques in the optimum design of real size pin jointed structures.
Comput Struct 87(5–6):284–302
10. Kaveh A, Talatahari S (2010) A discrete Big Bang–Big Crunch algorithm for optimal design of
skeletal structures. Asian J Civil Eng 11(1):103–122
11. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
12. American Institute of Steel Construction (AISC) (2001) Manual of steel construction: load and
resistance factor design. American Institute of Steel Construction, Chicago, IL
13. Dumonteil P (1992) Simple equations for effective length factors. AISC Eng J 29(3):111–115
14. Kaveh A, Talatahari S (2009) Hybrid algorithm of harmony search, particle swarm and ant
colony for structural design optimization. Stud Comput Intell 239:159–198
15. Kaveh A, Talatahari S (2012) Charged system search for optimal design of frame structures.
Appl Soft Comput 12:382–393
16. Camp CV, Bichon BJ, Stovall S (2005) Design of steel frames using ant colony optimization. J
Struct Eng ASCE 131:369–379
17. Degertekin SO (2012) Improved harmony search algorithms for sizing optimization of truss
structures. Comput Struct 92–93:229–241
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Chapter 5
Size and Geometry Optimization of
Double-Layer Grids Using the CBO
and ECBO Algorithms
5.1 Introduction
Space structures have become popular not only because of their topological attrac-
tiveness and greater reserves of strength compared to conventional structures but
also because of their easy and fast construction. Double-layer grids are ideally
suited for covering exhibition pavilions, assembly halls, swimming pools, hangars,
churches, bridge decks, and many types of industrial buildings in which large
unobstructed areas are required. Double-layer grids have been built successfully
at a lower cost than equivalent conventional systems, providing at the same time
additional advantages, such as greater rigidity, erection simplicity, and possibility
of covering larger areas.
These grids can be thought of as logical extensions of single-layer grid frame-
works, consisting of two or more sets of parallel beams intersecting each other at
right or oblique angles and loaded by forces perpendicular to the plane of the
framework. Single-layer grids are used for clear spans up to 10 m. For larger
spans, double-layer grids are more suitable and provide an economical solution
for spans up to 100 m. Double-layer grids consist of two plane grids (which are not
necessarily of identical layout) forming the top and bottom layers, parallel to each
other, and interconnected by vertical or inclined “web” diagonal members. Single-
layer grids are mainly under the action of flexural moments, whereas the component
members of double-layer grids are almost exclusively under the action of axial
forces. The elimination of bending moments leads to a full utilization of strength of
all the elements. Double-layer grids have a greater number of structural elements
and employing optimization techniques has a considerable impact on the economy
and efficient design of such structures [1]. This study focuses on economical
comparison of two commonly used double-layer grid configurations, namely,
two-way on two-way grid and diagonal on diagonal grid and determining their
optimum span-depth ratio. The span ranges of 15 15 m and 40 40 m with
certain bays of equal length in two directions are considered as small and big size
grids, respectively. Bottom layer is simply supported at the corner nodes and as
mid-edge at two parallel sides of the grid for the small and big span cases,
respectively. The discrete values of depth are chosen from a certain interval with
a 0.5 m increment for both cases to achieve the optimum value. For determining the
grouping effects, various grouping patterns are applied in each case. Finally the
20 20 m square on larger square grid for the effect of support location on the
weight of the double-layer grid is introduced. The discrete values of depth are
selected from a certain interval with a 0.25 m increment in this case [2].
Colliding bodies optimization (CBO) is a new metaheuristic search algorithm
that is developed by Kaveh and Mahdavi [3]. CBO is based on the governing laws
of one-dimensional collision between two bodies from the physics where an object
collides with another and they move toward the minimum energy level. The CBO is
simple in concept, depends on no internal parameters, and does not use memory for
saving the best-so-far solutions. The enhanced colliding bodies optimization
(ECBO) is introduced by Kaveh and Ilchi Ghazaan [4], and it uses memory to
save some historically best solutions to improve the CBO performance without
increasing the computational cost. In this method, some components of agents are
also changed to help the agents to escape from local minima. In this chapter, the
ability of the CBO and ECBO on optimal design of double-layer grids is examined
to carry out a precise comparison between different configurations. The design
algorithm is supposed to obtain minimum weight grid through suitable selection of
tube sections available in AISC-LRFD [5]. Strength constraints of AISC-LRFD
specifications and displacement constraints are imposed on grids. Moreover, three
other powerful advanced algorithms consisting of the HPSACO [6] (based on PSO,
ACO, and HS algorithms), the HBB–BC [7] (based on BB–BC and PSO methods),
and the CS [8] are applied to carry out a precise assessment and demonstrate the
effectiveness and robustness of the CBO and ECBO algorithms in achieving better
designs and estimating better depth for each type. Finally the effect of support
location on the weight of different kinds of double-layer grids is investigated using
ECBO algorithm.
The remainder of this chapter is organized as follows: In Sect. 5.2, the mathe-
matical formulation of the structural optimization problems is presented and a brief
explanation of the AISC-LRFD is provided. Section 5.3 includes an explanation of
the CBO and ECBO algorithms. In Sect. 5.4 structural models are explained and
three numerical examples are presented in Sect. 5.5. The last section concludes the
chapter.
The allowable cross sections are considered as 37 steel pipe sections as shown in
Table 5.1, where the abbreviations ST, EST, and DEST stand for standard weight,
extra strong, and double extra strong, respectively. These sections are taken from
AISC-LRFD [5] and this code is also utilized for design.
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5.2 Optimal Design of Double-Layer Grids 73
Table 5.1 The allowable steel pipe sections taken from AISC-LRFD
Nominal Weight per Gyration
Type diameter (in) ft (lb) Area (in2) I (in4) radius (in) J (in4)
1 ST ½ 0.85 0.25 0.017 0.261 0.034
2 EST ½ 1.09 0.32 0.02 0.25 0.04
3 ST ¾ 1.13 0.333 0.037 0.334 0.074
4 EST ¾ 1.47 0.433 0.045 0.321 0.09
5 ST 1 1.68 0.494 0.087 0.421 0.175
6 EST 1 2.17 0.639 0.106 0.407 0.211
7 ST 1¼ 2.27 0.669 0.195 0.54 0.389
8 ST 1½ 2.72 0.799 0.31 0.623 0.62
9 EST 1¼ 3.00 0.881 0.242 0.524 0.484
10 ST 2 3.65 1.07 0.666 0.787 1.33
11 EST 1½ 3.63 1.07 0.391 0.605 0.782
12 EST 2 5.02 1.48 0.868 0.766 1.74
13 ST 2½ 5.79 1.7 1.53 0.947 3.06
14 ST 3 7.58 2.23 3.02 1.16 6.03
15 EST 2½ 7.66 2.25 1.92 0.924 3.85
16 DEST 2 9.03 2.66 1.31 0.703 2.62
17 ST 3½ 9.11 2.68 4.79 1.34 9.58
18 EST 3 10.25 3.02 3.89 1.14 8.13
19 ST 4 10.79 3.17 7.23 1.51 14.5
20 EST 3½ 12.50 3.68 6.28 1.31 12.6
21 DEST 2½ 13.69 4.03 2.87 0.844 5.74
22 ST 5 14.62 4.3 15.2 1.88 30.3
23 EST 4 14.98 4.41 9.61 1.48 19.2
24 DEST 3 18.58 5.47 5.99 1.05 12
25 ST 6 18.97 5.58 28.1 2.25 56.3
26 EST 5 20.78 6.11 20.7 1.84 41.3
27 DEST 4 27.54 8.1 15.3 1.37 30.6
28 ST 8 28.55 8.4 72.5 2.94 145
29 EST 6 28.57 8.4 40.5 2.19 81
30 DEST 5 38.59 11.3 33.6 1.72 67.3
31 ST 10 40.48 11.9 161 3.67 321
32 EST 8 43.39 12.8 106 2.88 211
33 ST 12 49.56 14.6 279 4.38 559
34 DEST 6 53.16 15.6 66.3 2.06 133
35 EST 10 54.74 16.1 212 3.63 424
36 EST 12 65.42 19.2 362 4.33 723
37 DEST 8 72.42 21.3 162 2.76 324
ST standard weight; EST extra strong; DEST double extra strong
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74 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
where {X} is the set of design variables, ng is the number of member groups in
structure (number of design variables), D is the cross-sectional areas available for
groups according to Table 5.1, W({X}) presents weight of the grid, nm(i) is the
number of members for the ith group, and ρj and Lj denote the material density and
the length for the jth member of the ith group, respectively.
The constraint conditions for grid structures are briefly explained in the
following:
Displacement constraints:
δi δmax , i ¼ 1, 2, . . . , nn ð5:2Þ
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5.3 CBO and ECBO Algorithms 75
X
nn X
nm
f cost ðfXgÞ ¼ ð1 þ E1 :vÞE2 W ðfXgÞ, v ¼ vid þ viσ þ viλ ð5:6Þ
i¼1 i¼1
where v is the constraint violation function and υdi , υσi , and υλi are constraint
violations for displacement, stress, and slenderness ratio, respectively. E1 and E2
are penalty function exponents which are selected considering the exploration and
exploitation rate of the search space. Here E1 is set to unity; E2 is selected in a way
that it decreases the penalties and reduces the cross-sectional areas. Thus, in the first
steps of the search process, E2 is set to 1 and it linearly increases to 3.
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76 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
where fit(i) represents the objective function value of the ith CB and n is the number
of colliding bodies. After sorting colliding bodies according to their objective
function values in an increasing order, two equal groups are created: (i) stationary
group and (ii) moving group (Fig. 5.2). Moving objects collide with stationary
objects to improve their positions and push stationary objects toward better posi-
tions. The velocities of the stationary and moving bodies before collision (vi) are
calculated by
n
vi ¼ 0, i ¼ 1, . . . , ð5:8Þ
2
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5.3 CBO and ECBO Algorithms 77
n n
vi ¼ xin2 xi , i ¼ þ 1, þ 2 . . . , n ð5:9Þ
2 2
where xi is the position vector of the ith CB. The velocity of stationary and moving
CBs after the collision (v0i ) is evaluated by
miþn2 þ εmiþn2 viþn2 n
v0i ¼ i ¼ 1, 2, . . . , ð5:10Þ
mi þ miþn2 2
mi εmin2 vi n n
v0i ¼ i ¼ þ 1, þ 2, . . . , n ð5:11Þ
mi þ min2 2 2
iter
ε¼1 ð5:12Þ
iter max
where ε is the coefficient of restitution (COR) and iter and itermax are the current
iteration number and the total number of iterations for optimization process,
respectively. New positions of each group are stated by the following formulas:
0 n
xinew ¼ xi þ rand∘vi , i ¼ 1, 2, . . . , ð5:13Þ
2
0 n
xinew ¼ xin2 þ rand∘vi , i¼ þ 1, . . . , n ð5:14Þ
2
where xinew , xi , and v0i are the new position, previous position, and the velocity after
the collision of the ith CB, respectively. rand is a random vector uniformly
distributed in the range of [1,1], and the sign “∘” denotes an element-by-element
multiplication.
In the ECBO, a memory that saves a number of historically best CBs is utilized to
improve the performance of the CBO and reduce the computational cost. Further-
more, ECBO changes some components of CBs randomly to prevent premature
convergence [4]. In this section, in order to introduce the ECBO algorithm, the
following steps should be taken.
5.3.2.1 Initialization
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78 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
where x0i is the initial solution vector of the ith CB, xmin and xmax are the minimum
and the maximum allowable limits vectors, and random is a random vector with
each component being in the interval [0,1].
5.3.2.2 Search
Step 1: The value of the mass for each CB is calculated by Eq. (5.7).
Step 2: Colliding memory (CM) is considered to save some historically best CB
vectors and their related mass and objective function values. The size of the CM
is taken as n/10 (n is the population size) in this chapter. In each iteration,
solution vectors that are saved in the CM are added to the population, and the
same number of the current worst CBs are deleted.
Step 3: CBs are sorted according to their objective function values in an increasing
order. To select the pairs of CBs for collision, they are divided into two equal
groups: (i) stationary group and (ii) moving group.
Step 4: The velocities of stationary and moving bodies before collision are evalu-
ated by Eqs. (5.8) and (5.9), respectively.
Step 5: The velocities of stationary and moving bodies after collision are calculated
by Eqs. (5.10) and (5.11), respectively.
Step 6: The new location of each CB is evaluated by Eqs. (5.13) or (5.14).
Step 7: A parameter like Pro within (0, 1) is introduced which specifies whether a
component of each CB must be changed or not. For each CB Pro is compared
with rni (i ¼ 1,2,. . ., n) which is a random number uniformly distributed within
(0, 1). If rni < Pro, one dimension of ith CB is selected randomly and its value is
regenerated by
xij ¼ xj, min þ random: xj, max xj, min ð5:16Þ
where xij is the jth variable of the ith CB. xj, min and xj, max are the lower and upper
bounds of the jth variable. In this chapter, the value of Pro is set to 0.3.
Step 1: After the predefined maximum evaluation number, the optimization process
is terminated [9].
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5.5 Numerical Examples 79
Two commonly used configurations for double-layer grids considered in this study
are two-way on two-way and diagonal on diagonal square grids [10]. Two span
values of 15 15 m and 40 40 m with certain bays of equal lengths in two
directions are considered as small and big size spans. Simply supported condition
is employed for bottom layer at the corner nodes and mid-edge at two parallel sides
for the small and big span cases, respectively. The discrete values of depth are
chosen from a certain interval with a 0.5 m increment for both cases to achieve the
optimum value. At last the 20 20 m square on larger square grid for the effect of
support locations on the weight of the double-layer grid is introduced. The discrete
values of depth are selected from a certain interval with a 0.25 m increment in this
case [2].
As mentioned before double-layer grids have a large number of structural
elements, and in order to simplify the design, they should be divided into some
groups. The element grouping of such design can be selected by designers in any
scheme or patterns, but if the members with the same behavior are placed in the
same group, the design becomes more efficient and economical (e.g., all members
in one group have the same stress ratios, approximately). To address this issue, the
SAP2000 toolbox for auto and fully stressed design could be used to select the
element grouping pattern at the preliminary stage of design considering the stress
ratios of the elements. However, the selection of such pattern can be based on
experiences, engineering judgment, or administrative constraints. In this chapter
three element grouping patterns, namely, GP1, GP2, and GP3, are introduced for
the purpose of practical fabrication and determining the grouping effects on the
different systems. Considering different sections of the top-layer, bottom-layer, and
diagonal elements leads to the first grouping type which is only applied to the
15 15 m span case with three design variables. In the second one, the top-layer,
bottom-layer, and diagonal elements are put into different groups in a diamond-like
manner around central node. The GP3 grouping pattern is the same as the second
one, but it is in a square form. The configuration, support locations, and element
grouping patterns of double-layer grids are shown in Fig. 5.3. Due to symmetry,
only a quarter of the 15 15 m span case is shown in this figure. The element
grouping in the form of GP2 is depicted by dark and light hatching.
The double-layer grids are assumed as ball jointed, with top-layer joints being
subjected to concentrated vertical loads corresponding to the uniformly distributed
load of magnitude 200 kg/m2. Stress and slenderness constraints [Eqs. (5.3), (5.4),
and (5.5)] according to AISC-LRFD provisions and displacement limitations of
span/600 are imposed on all the nodes in vertical direction. The modulus of
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80 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
Fig. 5.3 Topology, element grouping, support locations for different cases; (a) 15 15 m
two-way on two-way grid, (b) 15 15 m diagonal on diagonal grid, (c) 40 40 m two-way on
two-way grid, (d) 40 40 m diagonal on diagonal grid [11]
elasticity is considered as 205 kN/mm2, and the yield stress of steel is taken as
248.2 MPa.
In CBO and ECBO, a population of n ¼ 30 CBs is utilized, and the size of
colliding memory is considered as n/10 that is taken as 3 for ECBO. The predefined
maximum evaluation number is considered as 15,000 analyses for all examples.
Because of the stochastic nature of the algorithms, each example has been solved
5 times independently. In all problems, CBs are allowed to select discrete values
from the permissible list of cross sections (real numbers are rounded to the nearest
integer in each iteration). The algorithms are coded in MATLAB and the structures
are analyzed using the direct stiffness method.
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5.5 Numerical Examples 81
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82
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ECBO 7471.287 5927.232 5590.446 6187.411 4444.858 4448.896 5381.764 3794.836 4263.224
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Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
5.5 Numerical Examples 83
Fig. 5.4 The convergence history for the 15 15 m diagonal on diagonal grid (GP3 and layer
thickness ¼ 1.5 m)
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84 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
Fig. 5.5 Best results of ECBO for 15 15 m double-layer grids in each group type: (a) two-way
on two-way grid and (b) diagonal on diagonal grid
Fig. 5.6 Best results of ECBO for 40 40 m double-layer grids in each group type: (a) two-way
on two-way grid and (b) diagonal on diagonal grid
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Table 5.4 Performance comparison for the 40 40 m double-layer grids (kg)
Two-way on two-way grid
GP2 GP3
Height ¼ 3 m Height ¼ 3.5 m Height ¼ 4 m Height ¼ 2.5 m Height ¼ 3 m Height ¼ 3.5 m
5.5 Numerical Examples
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ECBO 99,741.612 92,861.835 99,073.395 87,533.094 79,132.887 78,337.836
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85
86 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
optimum designs for the two-way on two-way grid in GP2 and GP3 grouping
schemes are 36 % and 26 % lighter than those of the diagonal on diagonal cases,
respectively. It can be realized that two-way on two-way grid is a more suitable
form for big span cases with the same number of span divisions (without consid-
ering the number and complexity of joints). It is apparent from the table that CBO
has obtained better results compared to HBB–BC and HPSACO in all cases. It
could also be seen that the enhanced version (ECBO) is capable of finding the best
results in all cases expect for one. The robustness of ECBO in size and geometry
optimization of big span double-layer grids is also evident.
In this case the 20 20 m square of large square double layer grid consisting of 136
nodes and 440 members is considered as the last example. Each span is divided into
eight bays with equal lengths in each direction. There are some empty spaces in the
middle of the grid created by removing some of the bottom-layer members ((usually
in tension). The attached bracings of the square on square offset at a rectangular
pattern lead to a construction lighter than the usual type (Fig. 5.7). Due to the
addition of the openings, this system is more suitable when the architect intends to
provide more natural lights inside the building (skylights can be placed within the
openings). This system is usually selected for structures subjected to normal range
of loads. A uniformly distributed load of 200 kg/m2 is transmitted to the concen-
trated vertical loads which are assigned to the nodes of the top grid proportional to
their load-bearing area. Double-layer grids can be supported by steel or concrete
columns, load-bearing brickworks, or perimeter ring beams. The positions of
Fig. 5.7 (a) Element grouping for square on larger square double-layer grid. (b) Configuration
and various types of support location
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5.5 Numerical Examples 87
25000
case 1 case 2
20000
Optimum weight (kg)
15000
10000
5000
0
1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
Layer Thickness (m)
Fig. 5.8 Effect of support location on the weight of square on larger square double-layer grid and
best results of ECBO algorithm
supports are important, as it influences the stress distribution. Often the locations of
the supports are selected considering the functional requirements of the building.
Sometimes architectural considerations may have a major effect on the location of
the supports as well as in the shape of the supporting structure. For shape and size
optimization, the ECBO is considered as optimization method. The grouping
pattern leads to 17 design variables in a square-like manner and is introduced for
the practical fabrication. Due to symmetry, only a quarter of this configuration is
shown in Fig. 5.7a. The range of discrete depths of [3, 5] is considered with a
0.25 m increment to achieve the optimum depth. Figure 5.8 shows the obtained
optimum weight, depth of grid, and comparison of the results between two cases of
support locations. ECBO obtains an optimum height of 1.75 m for this type of grid.
If possible, support at the extreme edges of the grid should be avoided as this will
produce heavy forces in the directly loaded members. Support positions slightly in
board are preferred. Often cantilevers can be provided by a proper support location;
this leads to considerable reduction in forces and deflections. As a rule, cantilevers
have little effect on shearing forces and hence on the size of the diagonals, but
cantilevers of approximately 0.3 of the clear span will result in a structure that has
less deflections, uses less material, and leads to a more uniform stress distribution.
The forces in the lower layer are nearly twice as much as the upper layer; however,
since these members are in tension, they are obviously not susceptible to buckling.
Table 5.5 shows the optimum design variables and best weight that ECBO has
produced as lightest design.
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88 5 Size and Geometry Optimization of Double-Layer Grids Using the CBO and. . .
In this chapter, the CBO and ECBO algorithms are examined in the context of size
and geometry optimization of double-layer grids designed for minimum weight.
The CBO has simple structure and depends on no internal parameters and does not
use memory for saving the best-so-far solutions. In order to improve the exploration
capabilities of the CBO and to prevent premature convergence, a stochastic
approach is employed in ECBO that changes some components of CBs randomly.
Colliding memory is also utilized to save a number of the best-so-far solutions to
reduce the computational cost. In order to indicate the similarities and differences
between the characteristics of the CBO and ECBO algorithms, two types of double-
layer grids with various span lengths are considered. Grids are designed in accor-
dance with AISC-LRFD specifications and displacement constraints. In small span
cases, diagonal on diagonal grid with more connections and members is a suitable
form because of greater rigidity and other advantages like convenience and appeal-
ing features. For big span cases, two-way on two-way grids with fewer number of
members are better than diagonal on diagonal ones. In this type of space structures,
if the positions of supports are slightly in board, the weight of structure is decreased
considerably due to reduction in forces and deflections as it influences the stress
distribution and leads to using less material and results in lighter weight designs.
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References 89
CBO has gained better results in small span case than three well-known algorithms
(CS, HBB–BC, and HPSACO) with small differences and for big sizes has gained
better design than HBB–BC and HPSACO. ECBO has better performance in all
cases than other methods because of the reliability of search, solution accuracy, and
speed of convergence. Generally, comparison of the results with other robustness
and hybridized algorithms shows the suitability and efficiency of the proposed
algorithms.
References
1. Makowski ZS (1990) Analysis, design and construction of double-layer grids, 1st edn. CRC
Press, UK
2. Kaveh A, Moradveisi M (2016) Optimal design of double-layer barrel vaults using CBO and
ECBO algorithms. Iran J Sci Technol Trans Civil Eng. doi:10.1007/s40996-016-0021-4
3. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel metaheuristic method.
Comput Struct 139:18–27
4. Kaveh A, Ilchi Ghazaan M (2014) Computer codes for colliding bodies optimization and its
enhanced version. Int J Optim Civil Eng 4:321–332
5. American Institute of Steel Construction (AISC) (1994) Manual of steel construction load
resistance factor design, 2nd edn. AISC, Chicago, IL
6. Kaveh A, Talatahari S (2009) Particle swarm optimizer, ant colony strategy and harmony
search scheme hybridized for optimization of truss structures. Comput Struct 87:267–283
7. Kaveh A, Talatahari S (2009) Size optimization of space trusses using Big Bang–Big Crunch
algorithm. Comput Struct 87:1129–1140
8. Yang XS, Deb S (2010) Engineering optimization by Cuckoo search. Int J Math Model Numer
Optim 1:330–343
9. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies algorithm for truss optimization
with dynamic constraints. J Comput Civil Eng 10.1061/(ASCE)CP.1943-5487.
0000445,04014104
10. Makowski ZS (1990) Analysis, design and construction of double-layer grids. Applied Science
Publisher Ltd, London
11. Kaveh A, Servati H (2002) Neural networks for the approximate analysis and design of double
layer grids. Int J Space Struct 17:77–89
12. Kaveh A, Bakhshpoori T, Afshari E (2011) An optimization-based comparative study of
double layer grids with two different configurations using cuckoo search algorithm. Int J
Optim Civil Eng 1:507–520
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Chapter 6
Sizing and Geometry Optimization
of Different Mechanical Systems of Domes via
the ECBO Algorithm
6.1 Introduction
This chapter deals with the optimal design of double-layer Lamella domes, suspen-
domes, and single-layer domes with relatively long spans including nonlinear
structural behavior [1]. In recent years, much progress has been made in the optimal
design of space structures by focusing on their linear behavior, neglecting non-
linearities which can result in uneconomic designs. In this study, geometric
nonlinearity optimization is taken into account for the abovementioned domes.
There are two main steps involved in the optimization of structural problems:
analysis and design. In this chapter, OPENSEES [2] is employed for analysis, and
enhanced colliding bodies is utilized in the design phase. All of the required
programs for the optimization phase are coded in MATLAB [3]. The design vari-
ables include cross-sectional areas of the structural elements, the height of dome,
the initial strain of cables, and the cross sections of cables in the suspen-dome. In
order to illustrate the efficiency of the proposed methodology, three numerical
examples including optimization of a single-layer dome with rigid joints, a
suspen-dome, and a double-layer dome with 12 rings subjected to dead and snow
loading are presented. The main contribution of the chapter is to utilize an efficient
metaheuristic algorithm for optimization of domes. Optimal design of structures is
usually achieved by considering the design variables to find an objective function
which is the minimum weight while all of the design constraints are satisfied.
The dome shape not only provides an elegant appearance but also offers one of
the most efficient interior environments for human residence because air and energy
circulation are managed without obstruction.
Suspen-Dome is a new style of prestressed space grid structure [4]. In recent
years, this type of dome has been used in some large-scale engineering structures,
such as Hikarigaoka Dome in Japan and Olympic Badminton Stadium of Beijing in
China. The symmetrical configuration of the Lamella dome and its triangular
configuration make it the topmost single-layer dome of the type. Figure 6.1
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6.2 Optimal Design Problem of Lamella Domes According to LRFD 93
In Sect. 6.4, three metaheuristic algorithms are compared for optimization of domes.
Comparative study is performed between optimal design of suspen-domes, single
layer with pin- and rigid-jointed domes, and double-layer Lamella domes using
ECBO algorithm in Sect. 6.5. Finally, Sect. 6.6 summarizes the main findings of
this chapter.
The allowable and standard cables which should be used in the tensegrity system
(hoop and radial cables) are shown in Table 6.1. The allowable cross sections of
steel elements, used in the domes, are standard 37 steel pipe sections shown in
Table 6.2. In this table, the abbreviations ST, EST, and DEST stand for standard
weight, extra strong, and double-extra strong, respectively. These sections are taken
from LRFD-AISC [14] which is also utilized as the code of practice. The process of
the optimal design of the dome structures includes introducing variables and
constraints and can be summarized as:
Find X ¼ x1 ; x2 ; ::; xng , h
xi 2 fd 1 , d 2 , . . . , dng
hi 2 fhmin , hmin þ h* , . . . , hmax
ð6:1Þ
To minimize
X nm
V ðxÞ ¼ xi :li
i¼1
δi δmax
i i ¼ 1, 2, . . . , nn: ð6:2Þ
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94 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Pu Mux Muy Pu
þ þ 1 for < 0:2 ð6:3Þ
2ϕc Pn ϕb Mnx ϕb Mny ϕc P n
Pu 8 Mux Muy Pu
þ þ 1 for 0:2 ð6:4Þ
ϕc Pn 9 ϕb Mnx ϕb Mny ϕ c Pn
where X is the vector containing the design variables of the elements; h is the crown
height; dj is the jth allowable discrete value for the design variables, hmin, hmax, and
h* are the permitted minimum, maximum, and increment values of the crown
height which in this chapter are taken as D/20, D/2, and 0.25 m, respectively, in
which D is the diameter of the dome; ng is the number of design variables or the
number of groups; V(x) is the volume of the structure; Li is the length of member i;
δi is the displacement of node i; δimax is the permitted displacement for the ith node;
nn is the total number of nodes; ϕc is the resistance factor (ϕc ¼ 0.9 for tension,
ϕc ¼ 0.85 for compression); ϕb is the flexural resistance reduction factor (ϕb ¼ 0.9);
Mux and Muy are the required flexural strengths in the x- and y-directions, respec-
tively; Mnx and Mny are the nominal flexural strengths in the x- and y-directions,
respectively; Pu is the required strength; and Pn denotes the nominal axial strength
which is computed as
Pn ¼ Ag Fcr ð6:5Þ
where k is the effective length factor taken as 1; l is the length of a dome member;
r is governing radius of gyration about the axis of buckling; and E is the modulus of
elasticity. In Eq. (6.9), Vu is the factored service load shear, Vn is the nominal
strength in shear, and φv represents the resistance factor for shear (φv ¼ 0.9).
V u φv V n ð6:9Þ
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6.2 Optimal Design Problem of Lamella Domes According to LRFD 95
Table 6.2 The allowable steel pipe sections taken from LRFD AISC
Nominal Weight per Area
Type diameter (in) ft. (lb) (in2) I (in4) S (in3) J (in4) Z (in3)
1 ST ½ 0.85 0.250 0.017 0.041 0.082 0.059
2 EST ½ 1.09 0.320 0.020 0.048 0.096 0.072
3 ST ¾ 1.13 0.333 0.037 0.071 0.142 0.100
4 EST ¾ 1.47 0.433 0.045 0.085 0.170 0.125
5 ST 1 1.68 0.494 0.087 0.133 0.266 0.187
6 EST 1 2.17 0.639 0.106 0.161 0.322 0.233
7 ST 1¼ 2.27 0.669 0.195 0.235 0.470 0.324
8 ST 1½ 2.72 0.799 0.310 0.326 0.652 0.448
9 EST 1¼ 3.00 0.881 0.242 0.291 0.582 0.414
10 EST 1½ 3.63 1.07 0.666 0.561 1.122 0.761
11 ST 2 3.65 1.07 0.391 0.412 0.824 0.581
12 EST 2 5.02 1.48 0.868 0.731 1.462 1.02
13 ST 2½ 5.79 1.70 1.53 1.06 2.12 1.45
14 ST 3 7.58 2.23 3.02 1.72 3.44 2.33
15 EST 2½ 7.66 2.25 1.92 1.34 2.68 1.87
16 DEST 2 9.03 2.66 1.31 1.10 2.2 1.67
17 ST 3½ 9.11 2.68 4.79 2.39 4.78 3.22
18 EST 3 10.25 3.02 3.89 2.23 4.46 3.08
19 ST 4 10.79 3.17 7.23 3.21 6.42 4.31
20 EST 3½ 12.50 3.68 6.28 3.14 6.28 4.32
21 DEST 2½ 13.69 4.03 2.87 2.00 4.00 3.04
22 ST 5 14.62 4.30 15.2 5.45 10.9 7.27
23 EST 4 14.98 4.41 9.61 4.27 8.54 5.85
24 DEST 3 18.58 5.47 5.99 3.42 6.84 5.12
25 ST 6 18.97 5.58 28.1 8.50 17.0 11.2
26 EST 5 20.78 6.11 20.7 7.43 14.86 10.1
27 DEST 4 27.54 8.10 15.3 6.79 13.58 9.97
28 ST 8 28.55 8.40 72.5 16.8 33.6 22.2
29 EST 6 28.57 8.40 40.5 12.2 24.4 16.6
30 DEST 5 38.59 11.3 33.6 12.1 24.2 17.5
31 ST 10 40.48 11.9 161 29.9 59.8 39.4
32 EST 8 43.39 12.8 106 24.5 49.0 33.0
33 ST 12 49.56 14.6 279 43.8 87.6 57.4
34 DEST 6 53.16 15.6 66.3 20.0 40.0 28.9
35 EST 10 54.74 16.1 212 39.4 78.8 52.6
36 EST 12 65.42 19.2 362 56.7 113.4 75.1
37 DEST 8 72.42 21.3 162 37.6 75.2 52.8
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96 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
P n ¼ F y Ag ð6:10Þ
Pn ¼ Fcr Ag ð6:11Þ
where Fcr is the critical stress based on flexural buckling of the member, calculated
using Eqs. (6.6) and (6.7).
In the above equations, l is the laterally unbraced length of the member, K is the
effective length factor, r is the governing radius of gyration about the axis of
buckling, and E is the modulus of elasticity.
This section introduces the enhanced colliding bodies optimization (ECBO) algo-
rithm. First, a brief description of standard CBO based on the work of Kaveh and
Mahdavi [15] is provided, and then the ECBO is introduced [16].
The collision is a natural occurrence, and the CBO algorithm was developed based
on this phenomenon. In this method, one object collides with the other object, and
they move toward a minimum energy level (Figure 6.2). The CBO is simple in
concept, does not depend on any internal parameter, and does not use memory for
saving the best-so-far solutions. CBO algorithm, like other multi-agent methods, is
a population-based metaheuristic algorithm. Each solution candidate Xi containing
a number of variables (i.e., Xi ¼ {xi,j}) is considered as a colliding body (CB). The
massed objects are divided into two equal groups, namely stationary and moving
objects, where moving objects collide with stationary objects to improve their
positions and push stationary objects toward better positions. After the collision,
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6.3 Metaheuristic Algorithm 97
(b)
V2f
(c) V1f
the new position of colliding bodies is updated based on the new velocity by using
the collision laws, and the lighter and heavier CBs move sharply and slowly,
respectively.
A modified version of the CBO which is presented by Kaveh and Mahdavi [15] is
ECBO, which improves the CBO to get faster and more reliable solutions [16]. The
introduction of a memory increases the convergence speed of ECBO with respect to
standard CBO. Furthermore, changing some components of colliding bodies will
help ECBO to escape from local optima. The steps involved in ECBO are as
follows:
Step 1: Initialization
The initial positions of all CBs are determined randomly in an m-dimensional
search space according to
where x0i is the initial solution vector of the ith CB. Here, xmin and xmax are the
bounds of design variables, rand is a random vector for which each component is
in the interval [0, 1], and n is the number of CBs.
Step 2: Defining mass
The value of mass for each CB is evaluated according to:
1
fitðkÞ
mk ¼ X n , k ¼ 1, 2, . . . , n ð6:13Þ
1
i¼1 fitðiÞ
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98 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Step 3: Saving
Considering a memory which saves some historically best CB vectors and their
related mass and objective function values can make the algorithm performance
better without increasing the computational cost [17]. Here a Colliding Memory
(CM) is utilized to save a number of the best-so-far solutions. Therefore in this
step, the solution vectors saved in CM are added to the population, and the same
number of current worst CBs are deleted. Finally, CBs are sorted according to
their masses in a decreasing order.
Step 4: Creating groups
CBs are divided into two equal groups: (i) stationary group and (ii) moving
group. The pairs of CBs are shown in Fig. 6.2.
Step 5: Criteria before the collision
The velocity of stationary bodies before collision is zero, i.e.,
n
vi ¼ 0, i ¼ 1, . . . , ð6:14Þ
2
Moving objects move toward stationary objects, and their velocities before
collision are calculated by
n
vi ¼ xin2 xi , i ¼ þ 1, . . . , n ð6:15Þ
2
miþn2 þ εmin2 viþn2 n
v0i ¼ i ¼ 1, 2, . . . , ð6:17Þ
mi þ miþn2 2
iter
ε¼1 ð6:18Þ
iter max
n
xinew ¼ xin2 þ rand 0 v0i , i¼ þ 1, . . . , n ð6:19Þ
2
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6.4 Configuration of Single-Layer Lamella Dome, Suspen-Dome, and Double-Layer Dome 99
n
xinew ¼ xi þ rand 0 v0i , i ¼ 1, 2, . . . , ð6:20Þ
2
where xij is the jth variable of the ith CB. xj,min and xj,max are the lower and upper
bounds of the jth variable, respectively. In order to protect the structures of CBs,
only one dimension is changed. This mechanism provides opportunities for the
CBs to move all over the search space, thus providing better diversity.
Step 9: Terminating condition check
The optimization process is terminated after a fixed number of iterations. If this
criterion is not satisfied, go to Step 2 for a new round of iteration.
For further details, the reader may refer to Kaveh and Mahdavi [18].
Topology of a single-layer Lamella dome is shown in Fig. 6.3. For all domes,
including the Lamella dome, it is possible to generate the geometric structural data
if four parameters consisting of the diameter (D) of the dome, the total number of
rings, the total number of joints, and the height of the crown (h) are known. When
the geometry of a dome is formed according to mentioned parameters, the topology
of domes can be obtained. The topology contains the total number of members,
member incidences, and total number of joints of the domes. The distances between
the rings in the dome on the meridian line are generally made to be equal. It can be
easily seen from Fig. 6.4a and b that all the joints are located with equal distances
on a particular ring in both domes. The top joint which is the dome’s crown is
numbered as first joint (joint number 1). The first joint on the first ring is numbered
as joint 2 in each dome type. In Lamella dome, there is the same number of joints on
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100 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Fig. 6.3 Schematic of a Lamella dome. (a) Plan view and (b) side view
each ring. The joint numbers of all the other first joints of other rings are computed
from the following equation:
J r1 þ ðr 1Þ 10 ð6:22Þ
where r is the ring number and Jr1 is the first joint number of the first ring, namely
2 for Lamella dome. It is worthwhile to mention that all of the first joints of the
odd-numbered rings (ring 1 and ring 3) are located on the radius that makes angle
of 16 with the x-axis and, similarly, the first joints of the even-numbered rings
(ring 2) are located on the intersection points of that ring and the x-axis in Lamella
dome. First member is taken as one and connects joint 1 to joint 2 which makes an
angle of (360/Nn) with x-axis in Lamella dome. For the first ring group, the start
node for all elements is the joint number 1 and the end nodes are those on the first
ring. The start and end nodes of ring elements can be obtained using Eqs. (6.24) and
(6.25), and for other rings (2 and 3), this process is repeated and all the member
incidences are similar.
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6.4 Configuration of Single-Layer Lamella Dome, Suspen-Dome, and Double-Layer Dome 101
(a) Z
(b)
h y1 y2 y3
Zi y4
I 38
r=1 r=2 r=3 R-h R
Xi
49 D/2 D/2
Fig. 6.4 (a) Joint coordinates of single-layer Lamella dome and (b) side view coordinate
8 !!
>
> D 360 Xi1
>
> xi ¼ cos i 4nj 1
>
>
>
>
2Nr 4ni
>
>
j¼1
>
>
>
> !!
< X
i1
D 360
yi ¼ sin i 4nj 1 ni ¼ 1, 2, . . . , Nr 1; i : Joint number
>
> 2Nr 4ni
>
> j¼1
>
>
>
> sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
>
> ni 2 D2
>
> ¼ 2
ðR hÞ
: i
z R
4Nr2
ð6:23Þ
I ¼ 10 ðni 1Þ þ j þ 1
ð j ¼ 1, 2, 3, . . . , 9Þ; ni ¼ 1, 2, . . . , Nr 1 ð6:24Þ
J ¼ 10 ðni 1Þ þ j þ 2
I ¼ 10 ðni 1Þ þ 2
ni ¼ 1, 2, . . . , Nr 1 ð6:25Þ
J ¼ 10 ni þ 1
360
ai ¼ ð6:26Þ
2Nn
360
ai ¼ i jr , 1 ð6:27Þ
2Nn
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102 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
r is the ring number that joint i is placed on it and j is the first joint number on the
ring number r which is on the x-axis. The members group which is used in Tables is
mentioned in the following sentences. For Lamella domes, the ribbed members
between the crown and the first ring are group 1, the diagonal members between fist
ring and second ring are group 2, and the diagonal members between second ring
and third ring are group 3. The members on the first ring are group 4, and the
members on the second ring are group 5.
The lower tensegric system is detached from the upper single-layer dome as an
independent system. In the lower tensegric system, the strands and the vertical
struts are hinged in the joints. The tensegrity system is constructed of four rings of
hoop steel cables, radial steel cables, and struts at the lower part of model. The
cables are tension-only elements and the vertical struts are also compression
elements.
The upper single-layer Lamella dome is arranged as a triangle circular truss. The
struts which are the web members of suspen-dome and bending members that are
the elements of single-layer Lamella dome are circular standard steel tubes for
which the sections are listed in Table 6.1.
As it was mentioned before, the suspen-dome is constructed by combining
tensegrity system (cable-struts) and a single-layer reticulated dome. The configu-
ration of single-layer Lamella dome is explained in the previous part. As can be
seen from Fig. 6.5, the tensegrity system is constructed of hoop cable, radial cable,
and compression struts. The topology of tension-only cables, which are called
radial and hoop cables, is the same as the upper single-layer reticulated dome.
Therefore, the suitable configuration of tensegrity system depends on its upper
single-layer dome.
The suspen-dome which is discussed in this study uses the configuration of a
Lamella dome as the upper part. Therefore, the configuration of tensegrity system
should be obtained using the configuration of the Lamella dome. The current
tensegrity system is connected to the rings 3, 4, and 5 of a single-layer Lamella
dome by vertical struts elements.
Computation of x and y coordinates of a joint on tensegrity system requires
the angle between the line that connects the considered joint to the joint placed
at the crown of dome (joint number 1) and the x-axis. For Lamella suspen-dome,
the mentioned angle can be computed by Eqs. (6.26) and (6.27) for the odd- and
even-numbered rings, respectively.
Computation of z coordinates of a joint on tensegrity system can be obtained
using the following equation:
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6.4 Configuration of Single-Layer Lamella Dome, Suspen-Dome, and Double-Layer Dome 103
(a) (b)
(c)
Fig. 6.5 Configuration of the double-layer dome or the tensegrity part of the suspen-dome. (a)
Radial elements of the double-layer dome and suspen-dome. (b) Vertical elements of the double-
layer dome and suspen-dome. (c) Hoop elements of the double-layer dome and suspen-dome
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
n2i D2
zi ¼ R
2
ðR hÞ HhoopðiÞ ð6:28Þ
4Nr 2
where Hhoop is the distance between the upper single-layer Lamella dome. In other
words, at the same time, it is the length of struts in the tensegrity structure.
For Lamella suspen-dome, the diagonal members between the crown and the
first ring are group 1, the diagonal members between first ring and the second ring
are group 2, the diagonal members between second ring and third ring are group
3, and the first ring, second ring, and third ring are groups 4, 5, and 6, respectively.
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104 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Then, after third ring each diagonal member and its related ring are numbered,
respectively. For example, if group 7 is the diagonal member between the ring 3 and
4, then the group 8 is the fourth ring of the dome.
The lower grid system is detached from the upper single-layer dome as an inde-
pendent system. In the lower system, the steel elements and the vertical struts are
hinged in the joints. The lower layer is constructed of four rings of hoop steel
elements, radial steel elements, and vertical elements at the lower part of model
where these can be subjected to tension and pressure, contrary to the cables\strands
in suspen-dome.
In double-layer domes, the upper single-layer Lamella dome is arranged as a
triangle circular truss. The vertical elements which are the web members of the
double-layer dome and bending members which are the elements of single-layer
Lamella dome are circular standard steel tubes.
As it mentioned, the double-layer dome is constructed by combining two layers
of the grids which are lower grid (steel tube strut) and single-layer reticulated dome.
The configuration of a single-layer Lamella dome is explained in previous part. As
can be seen from Fig. 6.5, the configuration of a double-layer dome is chosen
exactly the same as a suspen-dome.
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6.5 Convergence Curves of the Metaheuristic Algorithms 105
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106 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
X
nm
V ðXÞ ¼ A Li ð6:29Þ
i¼1
In this section, a single-layer Lamella dome is optimized using the ECBO algorithm
(Fig. 6.7). In this case, the dead and snow loads are considered for Lamella domes
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6.6 Comparison of Different Mechanical Systems of Domes 107
to investigate the real behavior and to obtain optimum geometry of dome under
these loading conditions. The dome structure is subjected to 0.8 kN/m2 of dead
load, 0.2 kN/m2 of live load, and 0.2 kN/m2 of basic wind pressure.
The number of rings is considered as 6 under this loading condition. The results
of the design are shown in Table 6.4. Due to the existence of a noticeable value of
dead/snow loading on each joint, the cross sections are obtained close to each other.
As can be seen, the optimal design of dome is found obtaining 5 m height for the
single-layer dome. For the dome with lower number of rings and lower number of
nodes, because of having the least number of joints and considerable amount of load
value on each joint, higher volume for dome is obtained and higher height is chosen
to provide higher stability. Because of this reason, the number of joints on each ring
in this study is chosen equal to 12. Also when the number of joints is increased, the
dead and snow loads are distributed among more joints.
The six-ring suspen-dome is employed as an example to illustrate this idea. The top
part of the model is a single-layer lattice dome, which has 6 rings with 12 joints in
each ring. The single-layer Lamella dome which is a popular type of latticed dome
consists of steel tube beams that are fixed at both ends to suspen-dome with rigid-
jointed topmost layer and steel tube trusses for suspen-dome with rigid-jointed
topmost layer.
Its design tensile strength is 240 MPa. The computational model is a suspen-
dome having a span of 40 m. The material of cables is made of high strength wire,
the technical parameters of these are provided in Table 6.1. These dome structures
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108 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Table 6.4 Optimal design of single-layer Lamella dome with rigid-jointed connections for
Lamella dome using ECBO algorithm
ECBO algorithm
Section
Number of rings 6
Optimum tubular Group 1 PIPST (8)
Section designations Group 2 PIPST (8)
Group 3 PIPST (8)
Group 4 PIPST (8)
Group 5 PIPST (8)
Group 6 PIPST (8)
Group 7 PIPST (8)
Group 8 PIPST (8)
Group 9 PIPST (6)
Group 10 PIPST (6)
Group 11 PIPST (6)
Group 12 PIPST (6)
Height of crown (m) 5.00
Maximum displacement (cm) 2.75
P
lI ðmÞ 982.52
Ā (cm2) 49.34
Maximum strength ratio 27.01
Volume (m3) 2.14
are subjected to 0.8 kN/m2 of dead load, 0.2 kN/m2 of live load, and 0.2 kN/m2 of
basic wind pressure.
In construction of the suspen-dome, the tensegrity system is constructed of three
rings of hoop cables, radial cables, and struts at the lower part of model. The
tensegrity system is connected to the rings 3, 4, and 5 of a single-layer Lamella
dome by vertical struts elements. For example, the struts of group 1 are connected to
the joints which are located in the third ring of the single-layer Lamella dome. The
struts are compression elements and have hinged connections on both ends; their
sections are circular steel tubes.
The tensegrity system is constructed of cables and struts stiffening the suspen-
dome structure (Figure 6.8). This also helps the suspen-dome to work like a double-
layer dome. Therefore, it is interesting to compare the optimum results of suspen-
dome with double-layer dome which is discussed in this study.
It is worthwhile to mention that the applied optimum prestressed force of
tensegrity system (radial and hoop cable) must be large enough to prevent cable
slack, but not so large as to make the struts buckle or induce very large opposite
moment compared to moment induced by external loads.
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6.6 Comparison of Different Mechanical Systems of Domes 109
As it was mentioned in the previous sections, the domes having rigid connections
are often used in the construction of long span single-layer domes, because the load
capacity and stiffness of pin-connected single-layer domes is very low. However,
pin connections are often used in double-layer lattice domes, because the additional
layer can make a more stiff structure compared to single-layer latticed dome
structures. For this reason, double-layer dome is studied here to compare it with
other systems of domes like single-layer and suspen-domes.
It is logical to use pin-jointed connections in the construction of the double-layer
dome systems. The upper layer of dome is constructed as a single-layer Lamella
dome. The lower system which is the second layer of dome is constructed utilizing
radial and vertical elements which stiffen the single layer of the dome structure. The
stiffness is provided by the second layer of dome. On the other hand, the moment
that is induced by the external load is sustained by two layers of the dome. This also
shows that the maximum bending moment of a double-layer dome which is
balanced by two layers of dome is decreased. Therefore, using double-layer dome
has two advantages consisting of reducing the element stresses and joint displace-
ments of the structure.
The six-ring double-layer dome is employed as an example to compare the
results with those of the previous example (Fig. 6.9). The computational model is
a double-layer dome having a span of 40 m. The top part of the model is a single-
layer Lamella dome, which has 6 rings and 12 joints in each ring. Both single-layer
Lamella domes consist of steel tube beams that are hinged at both ends and
constructed of steel tube trusses for double-layer dome which are made of
pin-jointed connections.
For construction of the double-layer dome, the lower layer (second layer)
consists of three rings of hoop elements, radial elements, and vertical elements at
the lower part of model. The second layer of double-layer dome is connected to the
rings 3, 4, and 5 of the single-layer Lamella dome by vertical elements. For
example, the vertical elements of group 1 are connected to the joints which are
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110 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
located in the third ring of the single-layer Lamella dome. The vertical elements are
compression elements and are standard steel elements having hinged connections
on both ends; its sections are circular steel tubes; also the hoop elements and radial
elements are standard steel elements which induce the main difference between the
standard double-layer dome and suspen-dome discussed in the previous section.
Same as single-layer Lamella dome and suspen-dome, the double-layer dome
structure is subjected to 0.8 kN/m2 of dead load, 0.2 kN/m2 of live load, and
0.2 kN/m2 of basic wind pressure.
6.6.4 Results
In this case of loading, the wind, dead, and snow loads are applied on all the domes
and the diameter is 20 m. It can be seen from Table 6.5 that the capacity of elements
in the single-layer Lamella dome with rigid joints is approximately 27 % of the
capacity of material which shows that material is overdesigned or on the other hand
the stress ratio of material does not control the design and the displacements govern
the design of single-layer Lamella dome with rigid joints. The maximum value of
displacement for this dome is equal to 2.75 cm and nearly the same as 2.80 which is
the maximum allowable displacement value of design. Therefore, displacement
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6.6 Comparison of Different Mechanical Systems of Domes 111
Table 6.5 Optimal design of upper single-layer dome with pin-jointed connections for Lamella
suspen-dome using the ECBO algorithm
ECBO algorithm
Pin-jointed
Number of rings 6
Optimum tubular Group 1 PIPST (3)
Section designations Group 2 PIPST (8)
Group 3 PIPST (10)
Group 4 PIPST (10)
Group 5 PIPST (8)
Group 6 PIPST (10)
Group 7 PIPST (10)
Group 8 PIPST (8)
Group 9 PIPST (10)
Group 10 PIPST (8)
Group 11 PIPST (10)
Group 12 PIPST (8)
Height of crown (m) 4.50
Maximum displacement (cm) 2.79
P
lI ðmÞ 979.37
Ā (cm2) 54.64
Maximum strength ratio 44.16
Volume (m3) 2.46
constraints are more active than the stress constraints for suspen-domes. The
optimum volume of single-layer dome is obtained 2.14 m3. The optimum height,
the total length of elements, and average cross-sectional area of single-layer dome
are obtained 5, 982.52, and 54.64, respectively.
It can be seen from Table 6.6 that using the capacity of elements in suspen-dome
with rigid joints is approximately 27 % more than the suspen-dome with pin-jointed
connections. Displacement constraints are more active than the stress constraints
for suspen-domes. The length of the struts which are connected to rings number
3, 4, and 5 are obtained as 1.5, 1, and 0.5 for domes, respectively. Therefore, the
least area sections are obtained for struts elements. When the tensegrity systems of
suspen-domes are compared, according to their optimum geometry design, it can be
seen that the cable system of the suspen-dome with rigid-jointed upper layer is more
economical (Tables 6.6, 6.7, and 6.8).
When these suspen-domes are compared, it can be seen that the suspen-dome
with rigid-jointed topmost layer provides a lighter design. For example, the opti-
mum volumes of the topmost layers for the domes with pin-jointed and rigid-jointed
connections are 2.46 m3 and 1.86 m3, respectively, which clearly shows that in
suspen-domes, the topmost layer with pin-jointed connections is 24 % heavier than
the topmost layer with rigid-jointed connections.
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112 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Table 6.7 Optimal design of upper single-layer dome with pin-jointed and rigid-jointed connec-
tions for Lamella suspen-dome using the ECBO algorithm
ECBO algorithm
Rigid-jointed
Number of rings 6
Optimum tubular Group 1 PIPST (3)
Section designations Group 2 PIPST (8)
Group 3 PIPST (8)
Group 4 PIPST (8)
Group 5 PIPST (4)
Group 6 PIPST (8)
Group 7 PIPST (10)
Group 8 PIPST (5)
Group 9 PIPST (10)
Group 10 PIPST (8)
Group 11 PIPST (5)
Group 12 PIPST (4)
Height of crown (m) 3.50
Maximum displacement (cm) 2.54
P
lI ðmÞ 976.54
Ā (cm2) 39.28
Maximum strength ratio 75.07
Volume (m3) 1.86
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6.6 Comparison of Different Mechanical Systems of Domes 113
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114 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
Table 6.9 Optimal design of double-layer dome with pin-jointed connections for Lamella suspen-
dome using ECBO algorithm
ECBO algorithm
Rigid-jointed
Number of rings 6
Optimum tubular Group 1 PIPST (3)
Section designations Group 2 PIPST (6)
Group 3 PIPST (6)
Group 4 PIPST (8)
Group 5 PIPST (6)
Group 6 PIPST (10)
Group 7 PIPST (10)
Group 8 PIPST (10)
Group 9 PIPST (10)
Group 10 PIPST (10)
Group 11 PIPST (10)
Group 12 PIPST (10)
Group 13 PIPST (8)
Group 14 PIPST (8)
Group 15 PIPST (8)
Group 16 PIPST (10)
Group 17 PIPST (8)
Group 18 PIPST (6)
Group 19 PIPST (8)
Group 20 PIPST (8)
Group 21 PIPST (6)
Height of crown (m) 5.00
Maximum displacement (cm) 2.80
P
lI ðmÞ 1577.04
Ā (cm2) 56.57
Maximum strength ratio 38.54
Volume (m3) 2.11
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6.7 Concluding Remarks 115
In this chapter, the ECBO is utilized for optimal design of different mechanical
systems of domes with pin and rigid-jointed connections. The different mechanical
systems of domes contain single-layer dome, double-layer dome and suspen-dome
with pin and rigid-jointed connections. The height of the domes, the length of the
strut/vertical elements, cables’ initial strain, the cross-sectional areas of the cables,
and the cross-sectional area of steel members are considered as design variables and
the volume of the entire structure is taken as the objective function. The optimiza-
tion method used in the chapter is based on the enhanced colliding bodies optimi-
zation algorithm. In this chapter, sizing and geometry of domes is presented. For
sizing optimization, the optimum steel section designations for the members of
domes are chosen from Table 6.2 and implemented in the design constraints from
LRFD-AISC.
A simple approach is presented to calculate the joint coordinates and specify the
elements to determine the configuration of single-layer Lamella domes and the
corresponding suspen-domes which are spatial prestressed structures with complex
mechanical characteristics. First, the joint coordinates are calculated, and then
using some simple relationships, the steel elements, struts, and cables are
constructed. This method considers not only the strength of steel components and
cables for optimal design as constraints but also considers the stability of the steel
members and controls the displacements of the overall structure.
An investigation on the efficiency of the ECBO method in optimal design of
single-layer domes is performed. In the suspen-dome structure and double-layer
dome, the tensegrity system and second layer significantly reduced the stresses and
the displacements of dome structure, respectively. By using the tensegrity system,
the suspen-dome structure performs like a double-layer dome structure. Therefore,
it is logical to use pin-connected joints in the construction of the suspen-dome
systems, and it is essential to compare them under the same conditions of loading as
discussed in this chapter. However, it is seen that the suspen-dome with upper layer
rigid joints offers a more economical design.
The ECBO method which is one of the recent additions to stochastic search
techniques of numerical optimization, is used to obtain the solution of the numerical
examples. It can be seen that the design examples of this study and the enhanced
colliding body method can be used for finding the solution of geometry and sizing
optimization of different mechanical system of domes such as double-layer domes,
suspend-domes which has complex mechanical structure, and single-layer domes.
As the future work, the cost of joints can also be added to the optimization
formulations. A comparative study can also be performed for other types of double-
layer and suspen-domes that are not studied in this chapter. Also optimum dynamic
analysis and design of different types of domes can be compared under seismic
loads.
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116 6 Sizing and Geometry Optimization of Different Mechanical Systems of Domes. . .
References
1. Kaveh A, Rezaei M (2016) Sizing and geometry optimization of different mechanical systems
of domes via enhanced colliding body algorithm. Neural Comput Appl (submitted)
2. McKenna F, Fenves G (2001) The opensees command language manual. University of
California, Berkeley (opensees.ce.berkeley.edu)
3. The Language of Technical Computing (2006) MATLAB. Math Works Inc., Natick, MA
4. Kawaguchi M, Abe M, Tatemichi I (1999) Design, tests and realization of suspen-dome
system. J Int Assoc Shell Spatial Struct 40(3):179–192
5. Kitipornchai S, Wenjiang K, Lam H-F, Albermani F (2005) Factors affecting the design and
construction of Lamella suspen-dome systems. J Constr Steel Res 61:764–785
6. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 39:18–27
7. Kaveh A, Talatahari S (2009) Size optimization of space trusses using Big Bang–Big Crunch
algorithm. Comput Struct 87(17–18):1129–1140
8. Saka MP (2007) Optimum geometry design of geodesic domes using harmony search algo-
rithm. Adv Struct Eng 10(6):595–606
9. Kaveh A, Talatahari S (2010) Optimal design of Schwedler and ribbed domes via hybrid Big
Bang–Big Crunch algorithm. J Constr Steel Res 66:412–419
10. Nie GB, Fan F, Zhi XD (2013) Test on the suspended dome structure and joints of Dalian
Gymnasium. Adv Struct Eng 16(3):467–486
11. Kamyab R, Salajegheh E (2013) Size optimization of nonlinear scallop domes by an enhanced
particle swarm algorithm. Int J Civil Eng 11(2):77–89
12. Kaveh A, Rezaei M (2016) Topology and geometry optimization of single-layer domes
utilizing CBO and ECBO. Sci Iranica 23(2):535–547
13. Kaveh A, Rezaei (2015) Topology and geometry optimization of different types of domes
using ECBO. Adv Comput Des 1(1):1–25
14. American Institute of Steel Construction (AISC) (1989) Manual of steel construction allow-
able stress design, 9th edn. AISC, Chicago, IL
15. Kaveh A, Mahdavi VR (2015) Colliding bodies optimization; extensions and applications.
Springer, Wien
16. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Struct Eng 77:66–75
17. Kaveh A, Ilchi Ghazaan M (2014) Computer codes for colliding bodies optimization and its
enhanced version. Int J Optim Civil Eng 4(3):321–332
18. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization method for optimal design of
truss structures with continuous variables. Adv Struct Eng 70:1–12
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Chapter 7
Simultaneous Shape–Size Optimization
of Single-Layer Barrel Vaults Using
an Improved Magnetic Charged System
Search Algorithm
7.1 Introduction
The use of braced barrel vaults as a lightweight space structures is very common
and it is worthwhile to investigate their optimal design [1]. Metaheuristic
algorithms explore the feasible region of the search space based on randomization
and some specified rules through a group of search agents. Nature-inspired phe-
nomena are commonly used as a basis for the rules employed by these agents [2].
In the field of size optimization of single-layer barrel vault frames, some studies
are carried out. Kaveh and Eftekhar have presented optimal design of barrel vault
frames using IBB–BC algorithm [3], in which a 173-bar single-layer barrel vault is
optimized under both symmetrical and unsymmetrical load cases. In a study by the
author and colleagues, size optimization of some single-layer barrel vault frames
via IMCSS algorithm [4] has been presented.
In a study carried out by Parke [5], several different configurations of braced
barrel vaults have been investigated using the stiffness method of analysis. Three
different configurations have been analyzed, each with five different span/height
ratios and under both cases of symmetrical and nonsymmetrical imposed nodal
loads. The reported study which was a comparative investigation demonstrates that
the most economical height-to-span ratio from weight point of view is approxi-
mately 0 17.
Some studies in the case of size optimization and a comparative study consid-
ering shape optimization are carried out for barrel vaults, but a more comprehensive
study of the problem of simultaneous shape–size optimization of these structures is
still needed. In this chapter, the latter problem is investigated using a new optimi-
zation approach. In this approach, a programming interface tool called OAPI is
utilized, and an improved version of a recently proposed algorithm called IMCSS
algorithm is used as the optimization tool.
Charged system search (CSS) is a relatively new metaheuristic optimization
algorithm proposed by Kaveh and Talatahari [6]. This algorithm is based on the
Coulomb and Gauss laws from physics and the governing laws of motion from the
Newtonian mechanics. The modified version of the CSS algorithm has also been
proposed by Kaveh et al. [2, 7]. In MCSS algorithm, the magnetic laws are also
considered in addition to electrical laws. In the present chapter, the IMCSS algo-
rithm is utilized. In the IMCSS algorithm, the harmony search scheme is used to
achieve better results. Some of the most effective parameters in the convergence
rate of algorithm are also modified.
This chapter is organized as follows: in Sect. 7.2, the problem of simultaneous
shape and size optimization for barrel vault frames is formulated. Section 7.3
presents the optimization approach. In Sect. 7.4, the static loading conditions acting
on the structures are defined. Two illustrative numerical examples are presented in
Sect. 7.5 to examine the efficiency of the proposed approach, and finally in
Sect. 7.6, the concluding remarks are derived.
where xi is the x coordinate of the ith joint, h is the height of barrel vault, and R is the
radius of semicircle which is expressed as
S2 þ 4h2
R¼ ð7:2Þ
8h
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7.2 Statement of Optimization Problem for Barrel Vault Frames 119
Find X ¼ ½x1 ; x2 ; x3 ; . . . ; xn , h
xi 2 fd1 , d2 , . . . , d37 g : Discrete Variables
hmin < h < hmax : Continous Variable ð7:3Þ
Shear constraint, for both major and minor axis (AISC-LRFD, Chapter G) [8]:
Vu
υis ¼ 1 0, i ¼ 1, 2, . . . , nm ð7:5Þ
φv V n
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120 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
where X is a vector which contains the design variables; for the discrete optimum
design problem, the variables xi are selected from an allowable set of discrete
values; n is the number of member groups; h is the height of barrel vault which is
known as the only shape variable; dj is the jth allowable discrete value for the
size design variables; hmin and hmax are the permitted minimum and maximum
values of the height which are, respectively, taken as S/20 and S/2 in this chapter;
S is the span of barrel vault; Mer(X) is the merit function; W(X) is the cost
function, which is taken as the weight of the structure; fpenalty(X) is the penalty
function which results from the violations of the constraints corresponding to the
response of the structure; nn is the number of nodes; δi and δi are the displace-
ment of the joints and the allowable displacement, respectively; nm is the number
of members; Vu is the required shear strength; Vn is the nominal shear strength
which is defined by the equations in Chap. G of the LRFD specification [8]; φv is
the shear resistance factor φv ¼ 0:9 ; Pu is the required strength (tension or
compression); Pn is the nominal axial strength (tension or compression); φc is
the resistance factor (φc ¼ 0:9 for tension, φc ¼ 0:85 for compression); Mu is the
required flexural strength, i.e., the moment due to the total factored load (sub-
script x or y denotes the axis about which bending occurs); Mn is the nominal
flexural strength determined in accordance with the appropriate equations in
Chap. F of the LRFD specification [8]; and φb is the flexural resistance reduction
factor (φb ¼ 0:9).
For the displacement limitations which must be considered to ensure the ser-
viceability requirements, the BS 5950 [9] limits the vertical deflections δv due to
unfactored loads to span/360, i.e., δV ¼ S=360 and horizontal displacements δH to
height/300, i.e., δH ¼ h=300 [10].
The nominal axial strength Pn is defined as
Pn ¼ Ag Fcr ð7:7Þ
where Fy is the specified minimum yield stress and the boundary between inelastic
and elastic instability is λc ¼ 1:5, where:
rffiffiffiffiffi
KL Fy
λc ¼ ð7:9Þ
rπ E
where K is the effective length factor for the member (K ¼ 1.0 for braced
frames [8]), L is the unbraced length of member, r is the governing radius of
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7.3 The Optimization Approach 121
gyration about plane of buckling, and E is the modulus of elasticity for the member
of structure.
The cost function can be expressed as
X
nm
W ðX Þ ¼ γ i xi Li ð7:10Þ
i¼1
X
nn Xnm
υk ¼ max υid ; 0 þ max υiI ; 0 þ max υis ; 0 ð7:12Þ
i¼1 i¼1
where υdi , υIi , υsi are the summation of displacement, shear, and interaction formula
penalties which are calculated by Eqs. (7.4) through (7.6), respectively.
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122 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Recently, the CSS algorithm and its modified version MCSS algorithm are, respec-
tively, presented by Kaveh and Talathari [6] and Kaveh et al. [7] for optimization
problems. The CSS algorithm takes its inspiration from the physical laws governing
a group of charged particles (CPs). These charged particles are sources of the
electric fields, and each CP can exert electric force on other CPs. The movement
of each CP due to the electric force can be determined using the Newtonian
mechanic laws. The MCSS algorithm considers the magnetic force in addition to
electric force for movement of CPs.
In this chapter, an improved version of MCSS algorithm called IMCSS is
presented. The IMCSS algorithm can be summarized as follows:
Level 1: Initialization
Step 1: Initialization. Initialize the algorithm parameters; the initial positions of CPs
are determined randomly in the search space
ð0Þ
xi, j ¼ xi, min þ rand ðxi:max xi, min Þ, i ¼ 1, 2, . . . , n: ð7:13Þ
ð0Þ
where xi;j determines the initial value of the ith variable for the jth CP; xi,min and
xi,max are the minimum and the maximum allowable values for the ith variable;
rand is a random number in the interval [0,1]; and n is the number of variables.
The initial velocities of charged particles are zero
ð0Þ
vi, j ¼ 0, i ¼ 1, 2, . . . , n: ð7:14Þ
fitðiÞ fitworst
qi ¼ , i ¼ 1, 2, . . . , N: ð7:15Þ
fitbest fitworst
where fitbest and fitworst are the best and the worst fitness of all particles; fit(i)
represents the fitness of the agent i; and N is the total number of CPs. The separation
distance rij between two charged particles is defined as
Xi Xj
r ij ¼
Xi þ Xj =2 Xbest
þ ε ; ð7:16Þ
where Xi and Xj are the positions of the ith and jth CPs, Xbest is the position of the
best current CP, and ε is a small positive number to avoid singularities.
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7.3 The Optimization Approach 123
Step 2. CP ranking. Evaluate the values of Merit function for the CPs, compare with
each other and sort them in an increasing order based on the corresponding value
of merit function.
Step 3. Creation of charged memory (CM). Store CMS number of the first CPs in
the CM.
Level 2: Search
Step 1: Force calculation. The probability of the attraction of the ith CP by the jth
CP is expressed as
8
< fitðiÞ fitbest
1 > rand or fitðjÞ > fitðiÞ,
pij ¼ fitðjÞ fitðiÞ ð7:17Þ
:
0 else:
where rand is a random number which is uniformly distributed in the range of (0,1).
The resultant electrical force FE,j acting on the jth CP can be calculated as follows:
FE , j ¼ q j
8
X q q
< i1 ¼ 1, i2 ¼ 0 , r ij < a,
i
r ij i1 þ i2 i2 pij Xi Xj , i ¼ 0, i2 ¼ 1 , r ij a,
a 3 r ij : 1
i, i6¼j j ¼ 1, 2, . . . , N:
ð7:18Þ
The probability of the magnetic influence (attracting or repelling) of the ith wire
(CP) on the jth CP is expressed as
1 fitðjÞ > fitðiÞ,
pmij ¼ ð7:19Þ
0 else:
where fit(i) and fit( j) are the objective values of the ith and jth CPs, respectively.
Such a definition ensures that only a good CP can affect a bad CP by the magnetic
force.
The resultant magnetic force FB,j acting on the jth CP due to the magnetic field of
the ith virtual wire (ith CP) can be expressed as
FB, j ¼ qj
8
X Ii Ii
< z1 ¼ 1, z2 ¼ 0 , r ij < R,
r z1 þ z2
2 ij
pmij Xi Xj , z ¼ 0, z2 ¼ 1 , r ij R,
R r ij : 1
i, i6¼j j ¼ 1, 2, . . . , N:
ð7:20Þ
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124 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
where qi is the charge of the ith CP, R is the radius of the virtual wires, Ii is the
average electric current in each wire, and pmij is the probability of the magnetic
influence (attracting or repelling) of the ith wire (CP) on the jth CP.
The average electric current in each wire Ii can be expressed as
df i, k df min, k
I avg ik ¼ sign df i, k ; ð7:21Þ
df max, k df min, k
where dfi,k is the variation of the objective function of the ith CP in the kth
movement (iteration). Here, fitk(i) and fitk1(i) are the values of the objective
function of the ith CP at the start of the kth and k 1th iterations, respectively.
Considering absolute values of dfi,k for all of the current CPs, dfmax,k and dfmin,k
would be the maximum and minimum values among these absolute values of df,
respectively.
A modification can be considered to avoid trapping in part of search space (Local
optima) because of attractive electrical force in CSS algorithm [7]:
F ¼ p r FE þ FB ; ð7:23Þ
where rand is a random number uniformly distributed in the range of (0,1), iter is
the current number of iterations, and itermax is the maximum number of iterations.
Step 2: Obtaining new solutions. Move each CP to the new position and calculate
the new velocity as follows:
Fj
Xj, new ¼ rand j1 ka Δt2 þ rand j2 kv V j, old Δt þ Xj, old ; ð7:25Þ
mj
where randj1 and randj2 are two random numbers uniformly distributed in the range
of (0,1). Here, mj is the mass of the jth CP which is equal to qj. Δt is the time step
and is set to unity. ka is the acceleration coefficient; kv is the velocity coefficient to
control the influence of the previous velocity. ka and kv are considered as
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7.3 The Optimization Approach 125
. .
ka ¼ c1 1 þ iter , kv ¼ c2 1 iter iter ; ð7:27Þ
itermax max
where c1 and c2 are two constants to control the exploitation and exploration of the
algorithm, respectively.
Step 3. Position correction of CPs. If each CP violates the boundary, its position is
corrected using an improved harmony search-based approach which is expressed
as follows:
In the process of position correction of CPs using harmony search-based
approach, the CMCR and PAR parameters help the algorithm to find globally
and locally improved solutions, respectively. PAR and bw in HS scheme are very
important parameters in fine-tuning of optimized solution vectors and can be
potentially useful in adjusting convergence rate of algorithm to reach better
solutions [13]. The standard version of CSS and MCSS algorithms use the
traditional HS scheme with constant values for both PAR and bw. Small PAR
values with large bw values can lead to poor performance of the algorithm and
considerable increase in iterations needed to find optimum solution. Although
small bw values in final iterations increase the fine-tuning of solution vectors, in
the first iterations bw must take a bigger value to enforce the algorithm to
increase the diversity of solution vectors. Furthermore, large PAR values with
small bw values usually lead to improvement of the best solutions in final
iterations and a better convergence to optimal solution vector. To improve the
performance of the HS scheme and eliminate the drawbacks which lie with
constant values of PAR and bw, the IMCSS algorithms use improved HS scheme
with the variable values of PAR and bw in position correction step. PAR and bw
change dynamically with iteration number as shown in Fig. 7.2 and are
expressed as follows [13]:
ðPARmax PARmin Þ
PARðiterÞ ¼ PARmin þ iter ð7:28Þ
itermax
and
(a) (b)
bw max
PAR max
PAR
bw
Fig. 7.2 Variation of (a) bw and (b) PAR versus iteration number in IMCSS algorithm [4]
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126 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
bw
Ln min =bwmax
c¼ ; ð7:30Þ
itermax
where PAR(iter) and bw(iter) are the values of PAR and bandwidth for current
iteration, respectively. bwmin and bwmax are the minimum and maximum band-
width, respectively.
Step 4: CP ranking. Evaluate and compare the values of merit function for the new
CPs, and sort them in an increasing order.
Step 5: CM updating. If some new CP vectors are better than the worst ones in the
CM (in terms of corresponding merit function), include the better vectors in the
CM and exclude the worst ones from the CM.
The present algorithms can be also applied to optimal design problems with discrete
variables. One way to solve discrete problems using a continuous algorithm is to
utilize a rounding function which changes the magnitude of a result to the nearest
discrete value as follows:
Fj
Xj, new ¼ Fix rand j1 ka Δt2 þ rand j2 kv V j, old Δt þ Xj, old ; ð7:31Þ
mj
where Fix(X) is a function which rounds each element of vector X to the nearest
permissible discrete value. Using this position updating formula, the agents will be
permitted to select discrete values [13].
Recently, Computers and Structures Inc. have introduced a powerful interface tool
known as Open Application Programming Interface (OAPI). The OAPI can be
utilized to automate and manage many of the processes required to build, analyze,
and design models through a programming language [14].
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7.4 Static Loading Conditions 127
According to ANSI-A58.1 [15] and ASCE/SEI 7-10 [16] codes, there are some
specific considerations for loading conditions of arched roofs such as barrel vault
structures. In this chapter, three static loading conditions are considered for opti-
mization of these structures which are expressed as follows:
A uniform dead load of 100 kg/m2 is considered for estimated weight of sheeting,
space frame, and nodes of barrel vault structure.
The snow load for arched roofs is calculated according to ANSI [15] and ASCE
[16] codes. Snow loads acting on a sloping surface shall be assumed to act on the
horizontal projection of that surface. The sloped roof (balanced) snow load, Ps,
shall be obtained by multiplying the flat roof snow load, Pf, by the roof slope factor,
Cs, as follows:
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128 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Ps ¼ Cs :Pf ð7:32Þ
where Cs is
8
< 15∘
< 1:0 α α15
>
Cs ¼ 1:0 15∘ < α < 60∘ ð7:33Þ
>
: 60
0:25 α > 60∘
The Cs distribution in arched roofs is shown in Fig. 7.3. In this chapter, the flat
roof snow load Pf is set to 150 kg/m2.
For wind load in arched roofs, different loads are applied in the windward quarter,
center half, and leeward quarter of the roof which are computed based on ANSI [15]
and ASCE [16] codes as
P ¼ q Gh Cp ð7:34Þ
where q is the wind velocity pressure, Gh is gust-effect factor, and Cp is the external
pressure coefficient. These parameters are calculated according to ANSI [15] and
ASCE [16] codes.
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7.5 Numerical Examples 129
This study presents optimal shape and size design of two single-layer barrel vault
frames which are first provided for size optimization by Kaveh et al. [4]. For all of
examples, a population of 100 charged particles is used, and the value of CMCR is
set to 0.95. The values of PARmin and PARmax in IMCSS algorithm are set to 0.35
and 0.9, respectively.
The two examples are discrete optimum design problems, and the variables are
selected from an allowable set of steel pipe sections taken from AISC-LRFD code
[17] shown in Table 7.1. For analysis of these structures, SAP2000 is used through
OAPI tool, and the optimization process is performed in MATLAB.
In all examples, the material density is 0.2836 lb/in3 (7850 kg/m3) and the
modulus of elasticity is 30,450 ksi (2.1 106 kg/cm2). The yield stress Fy of steel
is taken as 34,135.96 psi (2400 kg/cm2) for both problems.
The 173-bar single-layer barrel vault frame with a 2-way grid pattern is shown in
Fig. 7.4. This spatial structure consists of 108 joints and 173 members. There are
16 design variables in this problem which consist of size and shape variables. For
the process of size optimization, all members of this structure are categorized into
15 groups, as shown in Fig. 7.4b. Furthermore, for the problem of shape optimiza-
tion, the lower and upper bounds of height as the only shape variable are 1.5 m and
15 m, respectively. The nodal displacements are limited to 1.05 in (26 mm) in x,
y directions and 1.64 in (41 mm) in z direction.
The configuration of the 173-bar single-layer barrel vault is as follows:
• Span (S) ¼ 30 m (1181.1 in)
• Height (H ) ¼ 8 m (314.96 in)
• Length (L ) ¼ 30 m (1181.1 in)
According to ANSI/ASCE considerations mentioned in Sect. 7.4, this spatial
structure is subjected to three loading conditions:
A uniform dead load of 100 kg/m2 is applied on the roof. The applied snow and
wind loads on this structure are shown in Fig. 7.5a and b, respectively.
The convergence history for optimization of this structure using CSS, MCSS,
and IMCSS algorithms is shown in Fig. 7.6. Comparison of the optimal design
results using presented algorithms is also provided in Table 7.2.
As seen in Table 7.2, the IMCSS algorithm finds its best solutions in 89 iterations
(8900 analyses), but the CSS and MCSS algorithms have not found any better
solutions in 10,000 analyses. The best weight of IMCSS is 39,778.21 lb
(18,043.09 kg), while it is 41,589.25 lb and 42,957.98 lb for the MCSS and CSS
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Table 7.1 The allowable steel pipe sections taken from AISC-LRFD code [17]
130
Dimensions Properties
Nominal Weight per Moment of Elastic section Gyration Plastic section
Section name diameter (in.) ft (lb) Area (in.2) inertia (in.4) modulus (in.3) radius (in) modulus (in.3)
7
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21 XP10 2 5.02 1.48 0.868 0.731 0.766 1.02
22 XP12 2½ 7.66 2.25 1.92 1.34 0.924 1.87
23 XP2 3 10.25 3.02 3.89 2.23 1.14 3.08
24 XP2.5 3½ 12.5 3.68 6.28 3.14 1.31 4.32
Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
25 XP3 4 14.98 4.41 9.61 4.27 1.48 5.85
26 XP3.5 5 20.78 6.11 20.7 7.43 1.84 10.1
27 XP4 6 28.57 8.4 40.5 12.2 2.19 16.6
28 XP5 8 43.39 12.8 106 24.5 2.88 33
29 XP6 10 54.74 16.1 212 39.4 3.63 52.6
30 XP8 12 65.42 19.2 362 56.7 4.33 75.1
31 Double- XXP2 2 9.03 2.66 1.31 1.1 0.703 1.67
7.5 Numerical Examples
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131
132 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
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7.5 Numerical Examples 133
Fig. 7.5 The 173-bar single-layer barrel vault frame subjected to (a) snow and (b) wind loadings [4]
4
x 10
10 CSS
MCSS
9 IMCSS
8
Weight (lb.)
4
10 20 30 40 50 60 70 80 90 100
Iteration
Fig. 7.6 Convergence curves for the 173-bar single-layer barrel vault frame using CSS, MCSS,
and IMCSS algorithms
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134 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Table 7.2 Optimal solutions for simultaneous shape and size optimization of the 173-bar barrel
vault (in2)
CSS MCSS IMCSS
Design Area Area Area
variables Section name (in.2) Section name (in.2) Section name (in.2)
1 A1 ‘XP1’ 0.639 ‘XP1’ 0.639 P1 0.494
2 A2 ‘XP1.5’ 1.07 ‘XP1.25’ 0.881 P2.5 1.7
3 A3 ‘XXP2’ 2.66 ‘P2.5’ 1.7 XP1.5 1.07
4 A4 ‘P1.5’ 0.799 ‘XP2’ 1.48 P3 2.23
5 A5 ‘P3.5’ 2.68 ‘XP1.5’ 1.07 XP1.5 1.07
6 A6 ‘XP1.25’ 0.881 ‘P2.5’ 1.7 P1.5 0.799
7 A7 ‘XP2’ 1.48 ‘P1.5’ 0.799 P1 0.494
8 A8 ‘P10’ 11.9 ‘P10’ 11.9 P10 11.9
9 A9 ‘XP6’ 8.4 ‘XP6’ 8.4 XP6 8.4
10 A10 ‘XP6’ 8.4 ‘P10’ 11.9 XP6 8.4
11 A11 ‘P10’ 11.9 ‘XP6’ 8.4 P10 11.9
12 A12 ‘XP6’ 8.4 ‘P10’ 11.9 P10 11.9
13 A13 ‘XP6’ 8.4 ‘P6’ 5.58 P6 5.58
14 A14 ‘P6’ 5.58 ‘P6’ 5.58 P6 5.58
15 A15 ‘P12’ 14.6 ‘P10’ 11.9 XP6 8.4
16 Height 131.0308 in (3.33 m) 132.6162 in (3.37 m) 113.9046 in (2.89 m)
Weight. lb. 42,957.98 41,589.25 39,778.21
Weight. kg. 19,485.41 18,864.57 18,043.09
Max. displacement (in) 1.6118 1.4360 1.1277
Max. strength ratio 0.9865 0.9604 0.9516
No. of analyses 10,000 10,000 8900
close to ratio of 0.17 from Parke’s study. As seen in Table 7.2, the maximum
strength ratio for CSS, MCSS, and IMCSS algorithms is 0.9865, 0.9604, and
0.9516, respectively, and the maximum displacement is 1.6118 in, 1.4360 in, and
1.1277 in for the CSS, MCSS, and IMCSS algorithms, respectively.
Figure 7.7a–c provides strength ratios for all elements of the 173-bar single-
layer barrel vault frame for optimal results of CSS, MCSS, and IMCSS algorithms,
respectively. The figures show that all strength ratios of elements are lower than 1;
thus there is no violation of constraints in the optimal results of presented algo-
rithms, and all strength constraints are satisfied. The maximum strength ratios for
element groups of the 173-bar single-layer barrel vault frame are shown in Fig. 7.8a
through c for optimal results of the presented algorithms.
Table 7.3 provides a comparison for the results of present work on simultaneous
shape and size optimization with those of a previous study [15] on size optimization
of the 173-bar barrel vault. Comparison of best weight for both problems is also
shown in Table 7.4. As it can be seen in the results, the value of weight of structure
has been reduced by 14.59 %, 17.23 %, and 18.8 % via CSS, MCSS, and IMCSS
algorithms, respectively.
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7.5 Numerical Examples 135
(a) 1
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 173
Element Number
1
(b)
0.8
Strength Ratio
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 173
Element Number
(c) 1
0.8
Strength Ratio
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 173
Element Number
Fig. 7.7 Strength ratios for the elements of the 173-bar single-layer barrel vault frame for optimal
results of (a) CSS, (b) MCSS, and (c) IMCSS algorithms
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136 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
(a) 1
0.6
0.4
0.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Element Group
(b) 1
0.8
Max. Strength Ratio
0.6
0.4
0.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Element Group
(c) 1
0.8
Max. Strength Ratio
0.6
0.4
0.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Element Group
Fig. 7.8 Maximum strength ratios for element groups of the 173-bar single-layer barrel vault
frame for optimal results of (a) CSS, (b) MCSS, and (c) IMCSS algorithms
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7.5 Numerical Examples 137
Table 7.3 Comparison of the optimal solutions for the 173-bar single-layer barrel vault frame
Kaveh et al. [4] Present work
Simultaneous shape and size
Size optimization optimization
Design variables CSS MCSS IMCSS CSS MCSS IMCSS
1 A1 0.494 0.639 0.25 0.639 0.639 0.494
2 A2 0.494 0.433 0.25 1.07 0.881 1.7
3 A3 1.07 0.494 0.25 2.66 1.7 1.07
4 A4 0.333 0.333 0.25 0.799 1.48 2.23
5 A5 0.32 0.639 0.32 2.68 1.07 1.07
6 A6 0.881 1.07 0.32 0.881 1.7 0.799
7 A7 0.799 0.639 0.25 1.48 0.799 0.494
8 A8 11.9 11.9 14.6 11.9 11.9 11.9
9 A9 11.9 11.9 8.4 8.4 8.4 8.4
10 A10 11.9 11.9 11.9 8.4 11.9 8.4
11 A11 11.9 11.9 11.9 11.9 8.4 11.9
12 A12 11.9 11.9 11.9 8.4 11.9 11.9
13 A13 5.58 5.58 5.58 8.4 5.58 5.58
14 A14 5.58 5.58 5.58 5.58 5.58 5.58
15 A15 11.9 11.9 11.9 14.6 11.9 8.4
16 Height (in) Invariable Invariable Invariable 131.03 131.62 113.90
Weight (lb.) 50,295.90 50,247.66 48,985.05 42,957.98 41,589.25 39,778.21
Max. strength 0.8724 0.8689 0.8751 0.9865 0.9604 0.9516
ratio
No. of analyses 20,000 20,000 19,800 10,000 10,000 8900
Table 7.4 Comparison of the best weights for the 173-bar single-layer barrel vault frame
Best weight (lb.)
optimization problem CSS MCSS IMCSS
Size optimization [4] 50,295.90 50,247.66 48,985.05
Simultaneous shape and size optimization 42,957.98 41,589.25 39,778.21
Percent of reduction in best weights 14.59 % 17.23 % 18.80 %
This spatial structure which is shown in Fig. 7.9 has a three-way pattern [4]. The
structure consists of 117 joints and 292 members. The problem has 31 design
variables and consists of size and shape variables. In the problem of size optimiza-
tion, considering the symmetry of the geometry and loading conditions, all mem-
bers are grouped into 30 independent size variables as shown in Fig. 7.9b. For the
problem of shape optimization, the lower and upper bounds of height as the only
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138 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Fig. 7.9 The 292-bar single-layer barrel vault frame: (a) three-dimensional view, (b) member
groups in top view [4]
shape variable are 1.8 m and 18 m, respectively. The nodes are subjected to the
displacement limits of 1.31 in (33 mm) in x, y directions and 1.97 in (50 mm) in
z directions.
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7.5 Numerical Examples 139
Fig. 7.10 The 292-bar single-layer barrel vault frame subjected to (a) snow and (b) wind loadings [4]
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140 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Table 7.5 Optimal solutions for simultaneous shape and size optimization of the 292-bar barrel
vault (in2)
CSS MCSS IMCSS
Design Area Area Area
variables Section name (in.2) Section Name (in.2) Section name (in.2)
1 A1 ‘P12’ 14.6 ‘P10’ 11.9 P10 11.9
2 A2 ‘XP6’ 8.4 ‘XP6’ 8.4 P10 11.9
3 A3 ‘XP10’ 16.1 ‘XP8’ 12.8 XXP5 11.3
4 A4 ‘XXP5’ 11.3 ‘P10’ 11.9 XP6 8.4
5 A5 ‘XP6’ 8.4 ‘XP5’ 6.11 XP6 8.4
6 A6 ‘XP6’ 8.4 ‘XP6’ 8.4 XP6 8.4
7 A7 ‘XP6’ 8.4 ‘P10’ 11.9 P10 11.9
8 A8 ‘XXP5’ 11.3 ‘XP6’ 8.4 P10 11.9
9 A9 ‘XP6’ 8.4 ‘XXP5’ 11.3 P10 11.9
10 A10 ‘XP12’ 19.2 ‘P12’ 14.6 P12 14.6
11 A11 ‘XP2.5’ 2.25 ‘P1.25’ 0.669 XP3 3.02
12 A12 ‘XP3.5’ 3.68 ‘P2.5’ 1.7 P1 0.494
13 A13 ‘P2.5’ 1.7 ‘XXP3’ 5.47 XP1.5 1.07
14 A14 ‘P2.5’ 1.7 ‘P1.25’ 0.669 P1 0.494
15 A15 ‘XP2.5’ 2.25 ‘XP2.5’ 2.25 XP2.5 2.25
16 A16 ‘P2.5’ 1.7 ‘P2.5’ 1.7 XP3.5 3.68
17 A17 ‘P2.5’ 1.7 ‘XP5’ 6.11 P2.5 1.7
18 A18 ‘XP1.25’ 0.881 ‘P6’ 5.58 P1.5 0.799
19 A19 ‘XP3.5’ 3.68 ‘P2.5’ 1.7 P2.5 1.7
20 A20 ‘P0.75’ 0.333 ‘XP0.5’ 0.32 XP3 3.02
21 A21 ‘XP3’ 3.02 ‘P3’ 2.23 XP2 1.48
22 A22 ‘P4’ 3.17 ‘XP4’ 4.41 XP1.5 1.07
23 A23 ‘P2.5’ 1.7 ‘P2.5’ 1.7 XP1.5 1.07
24 A24 ‘P3’ 2.23 ‘P3’ 2.23 XP3 3.02
25 A25 ‘P2.5’ 1.7 ‘XP2’ 1.48 P3 2.23
26 A26 ‘P3’ 2.23 ‘XP2’ 1.48 P3 2.23
27 A27 ‘XP2.5’ 2.25 ‘XP4’ 4.41 XP3.5 3.68
28 A28 ‘P2.5’ 1.7 ‘XP3’ 3.02 P2.5 1.7
29 A29 ‘XP6’ 8.4 ‘XP2’ 1.48 P1.25 0.669
30 A30 ‘XP2.5’ 2.25 ‘XP2.5’ 2.25 XP1.25 0.881
31 Height 204.8791 in (5.20 m) 163.0436 in (4.14 m) 173.0666 in (4.40 m)
Weight. lb. 57,119.63 52,773.58 51,856.76
Weight. Kg. 25,909.03 23,937.69 23,521.83
Max. dis- 1.5802 1.5008 1.4424
placement
(in)
Max. strength 0.9413 0.9303 0.9746
ratio
No. of 13,200 12,500 12,200
analyses
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7.5 Numerical Examples 141
5
x 10
2
CSS
MCSS
IMCSS
1.5
Weight (lb)
0.5
20 40 60 80 100 120 140
Iteration
Fig. 7.11 Convergence history for the 292-bar single-layer barrel vault frame using CSS, MCSS,
and IMCSS algorithms
The best value for height of this barrel vault from CSS, MCSS, and IMCSS
algorithms is 204.88 in, 163.04 in, and 173.07 in, respectively. The best height-to-
span ratios, therefore, obtained from CSS, MCSS, and IMCSS algorithms are 0.15,
0.12, and 0.12, respectively, which are approximately close to value of 0.17 from
Parke’s study.
Table 7.5 also shows the maximum displacement and strength ratios for all
algorithms. The values of maximum strength ratio for CSS, MCSS, and IMCSS
algorithms are 0.9413, 0.9303, and 0.9746, respectively, and the values of maxi-
mum displacement are 1.5802 in. 1.5008 in. and 1.4424 in. respectively. The
strength ratios for all elements of the 292-bar single-layer barrel vault are depicted
in Fig. 7.12a through c, and the maximum strength ratios for element groups of this
structure are presented in Fig. 7.13a through c for optimal results of CSS, MCSS,
and IMCSS algorithms, respectively.
As shown in Fig. 7.12a–c, all of the strength ratios of elements are lower than 1;
therefore, all of the presented algorithms have no violation of constraints in their
best solutions, and the constraints are satisfied.
Table 7.6 draws a comparison between the results of present work on simulta-
neous shape and size optimization and those of a previous study on size optimiza-
tion [4] for this structure. On comparison of the best weights for presented
algorithms shown in Table 7.7, the value of weight of structure has decreased by
16.4 %, 17.23 %, and 17.65 % via CSS, MCSS, and IMCSS algorithms,
respectively.
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142 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
(a) 1
0.8
Strength Ratio
0.6
0.4
0.2
0
0 50 100 150 200 250 292
Element Number
(b) 1
0.8
Strength Ratio
0.6
0.4
0.2
0
0 50 100 150 200 250 292
Element Number
(c) 1
0.8
Strength Ratio
0.6
0.4
0.2
0
0 50 100 150 200 250 292
Element Number
Fig. 7.12 Strength ratios for the elements of the 292-bar single-layer barrel vault frame for
optimal results of (a) CSS, (b) MCSS, and (c) IMCSS algorithms
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7.5 Numerical Examples 143
1
(a)
0.8
Max. Strength Ratio
0.6
0.4
0.2
0
0 5 10 15 20 25 30
Element Group
(b) 1
0.8
Max. Strength Ratio
0.6
0.4
0.2
0
0 5 10 15 20 25 30
Element Group
(c) 1
0.8
Max. Strength Ratio
0.6
0.4
0.2
0
0 5 10 15 20 25 30
Element Group
Fig. 7.13 Maximum strength ratios for element groups of the 292-bar single-layer barrel vault
frame for optimal results of (a) CSS, (b) MCSS, and (c) IMCSS algorithms
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144 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
Table 7.6 Comparison of the optimal solutions for the 292-bar single-layer barrel vault frame
Kaveh et al. [4] Present work
Simultaneous shape and size
Size optimization optimization
Design variables CSS MCSS IMCSS CSS MCSS IMCSS
1 A1 14.6 14.6 14.6 14.6 11.9 11.9
2 A2 11.9 8.4 8.4 8.4 8.4 11.9
3 A3 12.8 12.8 11.9 16.1 12.8 11.3
4 A4 5.58 14.6 8.4 11.3 11.9 8.4
5 A5 12.8 11.9 11.9 8.4 6.11 8.4
6 A6 11.9 11.9 11.9 8.4 8.4 8.4
7 A7 11.9 14.6 11.9 8.4 11.9 11.9
8 A8 14.6 16.1 14.6 11.3 8.4 11.9
9 A9 11.9 11.9 11.9 8.4 11.3 11.9
10 A10 19.2 19.2 14.6 19.2 14.6 14.6
11 A11 2.25 0.25 1.48 2.25 0.669 3.02
12 A12 0.669 0.433 0.799 3.68 1.7 0.494
13 A13 6.11 1.7 0.669 1.7 5.47 1.07
14 A14 3.68 0.639 0.799 1.7 0.669 0.494
15 A15 1.7 0.669 0.494 2.25 2.25 2.25
16 A16 3.17 1.07 0.799 1.7 1.7 3.68
17 A17 1.48 2.68 2.25 1.7 6.11 1.7
18 A18 1.48 1.07 0.669 0.881 5.58 0.799
19 A19 5.47 0.639 0.639 3.68 1.7 1.7
20 A20 4.3 2.23 1.48 0.333 0.32 3.02
21 A21 2.66 1.48 0.799 3.02 2.23 1.48
22 A22 2.25 1.07 1.07 3.17 4.41 1.07
23 A23 0.639 2.23 0.799 1.7 1.7 1.07
24 A24 1.48 1.7 1.07 2.23 2.23 3.02
25 A25 0.799 0.669 0.669 1.7 1.48 2.23
26 A26 1.07 0.669 0.881 2.23 1.48 2.23
27 A27 0.799 1.7 0.799 2.25 4.41 3.68
28 A28 1.48 2.23 0.799 1.7 3.02 1.7
29 A29 1.07 0.799 1.48 8.4 1.48 0.669
30 A30 2.68 0.799 12.8 2.25 2.25 0.881
31 Height (in) Invariable Invariable Invariable 204.88 163.04 173.07
Weight (lb.) 68,324.57 65,892.33 62,968.19 57,119.63 52,773.58 51,856.76
Max. strength 0.9527 0.8883 0.9939 0.9413 0.9303 0.9746
ratio
No. of analyses 20,000 20,000 17,500 13,200 12,500 12,200
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References 145
Table 7.7 Comparison of the best weights for the 292-bar single-layer barrel vault frame
Best weight (lb.)
Optimization problem CSS MCSS IMCSS
Size optimization [4] 68,324.57 65,892.33 62,968.19
Simultaneous shape and size optimization 57,119.63 52,773.58 51,856.76
Percent of reduction in best weights 16.40 % 19.91 % 17.65 %
This chapter has applied an optimization approach which contains improved mag-
netic charged system search (IMCSS) and open application programming interface
(OAPI) for simultaneous shape and size optimization of barrel vault frames. In this
approach, OAPI is utilized as a programming interface tool through programming
language to manage the process of structural analysis during the optimization
process, and the IMCSS which is an improved version of MCSS algorithm is
used for achieving better solutions for the optimization problem.
Two single-layer barrel vault frames with different patterns are optimized via the
presented approach. In the process of optimization, contrary to size variables, shape
is a continuous variable. In the case of shape optimization of this type of space
structures, since all of the nodal coordinates of the shape variables are dependent on
the height-to-span ratio of the barrel vault, height is considered as the only shape
variable in a constant span of barrel vault.
In comparison, the best height-to-span ratios of barrel vaults under static loading
conditions obtained from CSS, MCSS, and IMCSS algorithms are approximately
close to value of 0.17 from comparative study carried out by Parke. Furthermore, as
seen in the results, different patterns of barrel vaults have different effects on the
value of best height-to-span ratio. Moreover, in comparison to CSS and MCSS
algorithms, IMCSS has found more optimal values for the weight of structures in a
lower number of analyses.
Since SAP2000 is a powerful software in modeling, analyzing, and designing of
large-scale spatial structures, OAPI would be a profit interface tool between this
software and MATLAB in the process of structural optimization.
References
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146 7 Simultaneous Shape–Size Optimization of Single-Layer Barrel Vaults Using. . .
4. Kaveh A, Mirzaei B, Jafarvand A (2013) Optimal design of single-layer barrel vault frames
using improved magnetic charged system search. Int J Optim Civil Eng 3(4):575–600
5. Parke GAR (2006) Comparison of the structural behaviour of various types of braced barrel
vaults. In: Makowski ZS (ed) Analysis, design and construction of braced barrel vaults. Taylor
& Francis e-Library, Hoboken, NJ, pp 113–144
6. Kaveh A, Talatahari S (2010) Optimal design of truss structures via the charged system search
algorithm. Struct Multidisip Optim 37(6):893–911
7. Kaveh A, Motie Share MA, Moslehi M (2013) Magnetic charged system search: a new
metaheuristic algorithm for optimization. Acta Mech 224(1):85–107
8. American Institute of Steel Construction (AISC) (2005) Construction manual, 13th edn. AISC,
Chicago, IL
9. British Standards Institution (BSI) (2000) Structural use of steelwork in building Part 1: code
of practice for design-rolled and welded section (BS5950-1). British Standards Institution,
London
10. Kameshki ES, Saka MP (2007) Optimum geometry design of nonlinear braced domes using
genetic algorithm. Comput Struct 85:71–79
11. Kaveh A, Farahmand Azar B, Talatahari S (2008) Ant colony optimization for design of space
trusses. Int J Space Struct 23(3):167–181
12. Kaveh A, Mirzaei B, Jafarvand A (2014) Optimal design of double layer barrel vaults using
improved magnetic charged system search. Asian J Civil Eng (BHRC) 15(1):135–154
13. Kaveh A, Bakhshpoori T, Ashoory M (2012) An efficient optimization procedure based
on cuckoo search algorithm for practical design of steel structures. Int J Optim Civil Eng
2(1):1–14
14. Computers and Structures Inc. (CSI) (2011) SAP2000 OAPI documentation. University of
California, Berkeley, CA
15. American National Standards Institute (ANSI) (1980) Minimum design loads for buildings and
other structures (ANSI A58.1)
16. American Society of Civil Engineers (ASCE) (2010) Minimum design loads for buildings and
other structures (ASCE-SEI 7-10)
17. American Institute of Steel Construction (AISC) (1994) Manual of steel construction—load
and resistance factor design (AISC-LRFD), 2nd edn. AISC, Chicago, IL
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Chapter 8
Optimal Design of Double-Layer Barrel
Vaults Using CBO and ECBO Algorithms
8.1 Introduction
Barrel vault is one of the oldest architectural forms, used since antiquity. The brick
architecture of the Orient or the masonry construction of the Romans provides
numerous examples of the structural use of barrel vaults. The industrial and
technological developments which have taken place during the last three decades
have had a far-reaching effect upon contemporary architecture and modern engi-
neering. New building techniques, new constructional materials, and new structural
forms have been introduced all over the world. The architectural search for new
structural forms has resulted in the widespread use of three-dimensional structures.
The evolution of effective computer techniques of analysis is undoubtedly one of
the reasons for the truly phenomenal acceptance of space structures. During recent
years, architects and engineers have rediscovered the advantages of barrel vaults as
viable and often highly suitable forms for covering not only low-cost industrial
buildings, warehouses, large-span hangars, and indoor sports stadiums but also
large cultural and leisure centers. The impact of industrialization on prefabricated
barrel vaults has proved to be the most significant factor leading to lower costs for
these structures. A barrel vault consists of one or more layers of elements that are
arched in one direction [1]. Barrel vaults are given different names depending on
the way their surface is formed. The earlier types of barrel vaults were constructed
as single-layer structures [2–4]. Nowadays, with increase of the spans, double-layer
systems are often preferred. Whereas the single-layer barrel vaults are mainly under
the action of flexural moments, the component members of double-layer barrel
vaults are almost exclusively under the action of axial forces; the elimination of
bending moments leads to a full utilization of strength of all the elements. Formex
algebra is a mathematical system that provides a convenient medium for configuration
processing. The concepts are general and can be used in many fields. In particular,
the ideas may be employed for generation of information about various aspects of
structural systems such as element connectivity, nodal coordinates, details of
loadings, joint numbers, and support arrangements. The information generated may
be used for various purposes, such as graphic visualization or input data for
structural analysis. Double-layer barrel vaults have great number of structural
elements, and utilizing optimization techniques has considerable influence on the
economy.
Methods of optimization can be divided into two general categories of gradient-
based methods and metaheuristic algorithms. Many of gradient-based optimization
algorithms have difficulties when dealing with large-scale optimization problems.
To overcome these difficulties, utilizing metaheuristic algorithms is inevitable. The
formulation of metaheuristic algorithms is often inspired by either natural phenom-
ena or physical laws. A metaheuristic algorithm consists of two phases: exploration
of the search space and exploitation of the best solutions found. One of the main
problems in developing a good metaheuristic algorithm is to maintain a reasonable
balance between the exploration and exploitation abilities. In the past decades,
structural optimization has been studied by using different metaheuristic algorithms
[5]. Colliding bodies optimization (CBO) is a new metaheuristic search algorithm
that is developed by Kaveh and Mahdavi [6]. CBO is based on the governing laws
of one-dimensional collision between two bodies in the physics that one object
collides with the other object and they move toward a minimum energy level. CBO
is simple in concept, depends on no internal parameters, and does not use memory
for saving the best-so-far solutions. The enhanced colliding bodies optimization
(ECBO) is introduced by Kaveh and Ilchi Ghazaan [7], and it uses memory to save
some historically best solutions to improve the CBO performance without increas-
ing the computational cost. In this method, some components of agents are also
changed to jump out from local minima. In this chapter, the performance of the
CBO and ECBO on optimal design of double-layer barrel vaults is examined. The
design algorithm is supposed to obtain minimum weight grid through suitable
selection of tube sections available in AISC-LRFD [8]. The strength and stability
requirements of steel members are imposed according to AISC-ASD [9].
The remainder of this chapter is organized as follows: In Sect. 8.2, the mathe-
matical formulation of the structural optimization problems is presented and a brief
explanation of the AISC-ASD is provided. Section 8.3 includes an explanation of
the CBO and ECBO algorithms. In Sect. 8.4 structural models are explained and
three numerical examples are presented. The last section concludes the chapter.
The allowable cross sections are considered as 37 steel pipe sections shown in
Table 8.1, where the abbreviations ST, EST, and DEST stand for standard weight,
extra strong, and double extra strong, respectively. These sections are taken from
AISC-LRFD [8] which is also utilized as the code of design.
The aim of optimizing the truss structures is to find a set of design variables that
has the minimum weight satisfying certain constraints. This can be expressed as
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8.2 Optimum Design of Double-Layer Barrel Vaults 149
Table 8.1 The allowable steel pipe sections taken from AISC-LRFD
Nominal Gyration
Type diameter (in) Weight per ft (lb) Area (in2) I (in4) radius (in) J (in4)
1 ST ½ 0.85 0.25 0.017 0.261 0.034
2 EST ½ 1.09 0.32 0.02 0.250 0.040
3 ST ¾ 1.13 0.333 0.037 0.334 0.074
4 EST ¾ 1.47 0.433 0.045 0.321 0.090
5 ST 1 1.68 0.494 0.087 0.421 0.175
6 EST 1 2.17 0.639 0.106 0.407 0.211
7 ST 1¼ 2.27 0.669 0.195 0.54 0.389
8 ST 1½ 2.72 0.799 0.31 0.623 0.620
9 EST 1¼ 3.00 0.881 0.242 0.524 0.484
10 ST 2 3.65 1.07 0.666 0.787 1.330
11 EST 1½ 3.63 1.07 0.391 0.605 0.782
12 EST 2 5.02 1.48 0.868 0.766 1.740
13 ST 2½ 5.79 1.7 1.53 0.947 3.060
14 ST 3 7.58 2.23 3.02 1.16 6.030
15 EST 2½ 7.66 2.25 1.92 0.924 3.850
16 DEST 2 9.03 2.66 1.31 0.703 2.620
17 ST 3½ 9.11 2.68 4.79 1.34 9.580
18 EST 3 10.25 3.02 3.89 1.14 8.130
19 ST 4 10.79 3.17 7.23 1.51 14.50
20 EST 3½ 12.50 3.68 6.28 1.31 12.60
21 DEST 2½ 13.69 4.03 2.87 0.844 5.740
22 ST 5 14.62 4.3 15.2 1.88 30.30
23 EST 4 14.98 4.41 9.61 1.48 19.20
24 DEST 3 18.58 5.47 5.99 1.05 12.00
25 ST 6 18.97 5.58 28.1 2.25 56.3
26 EST 5 20.78 6.11 20.7 1.84 41.3
27 DEST 4 27.54 8.1 15.3 1.37 30.6
28 ST 8 28.55 8.4 72.5 2.94 145
29 EST 6 28.57 8.4 40.5 2.19 81
30 DEST 5 38.59 11.3 33.6 1.72 67.3
31 ST 10 40.48 11.9 161 3.67 321
32 EST 8 43.39 12.8 106 2.88 211
33 ST 12 49.56 14.6 279 4.38 559
34 DEST 6 53.16 15.6 66.3 2.06 133
35 EST 10 54.74 16.1 212 3.63 424
36 EST 12 65.42 19.2 362 4.33 723
37 DEST 8 72.42 21.3 162 2.76 324
ST Standard weight; EST Extra strong; DEST Double extra strong
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150 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Find fXg ¼ x1 , x2 , x3 , . . . , xng , xi 2 D ¼ fd1 , d2 , d3 , . . . , d 37 g
Xng X
nm ðiÞ
ð8:1Þ
To minimize W ðfXgÞ ¼ xi ρj :Lj
i¼1 j¼1
where {X} is the set of design variables, ng is the number of member groups in
structure (number of design variables), D is the list of cross-sectional areas avail-
able for groups according to Table 8.1, W({X}) presents weight of the structure, nm
(i) is the number of members for the ith group, nn and ns are the number of nodes
and number of compression elements, respectively, σ i is the element stress and δi is
the nodal displacement, and ρj and Lj denote the material density and the length for
the jth member of the ith group, respectively. σ bi is the allowable buckling stress in
member i when it is in compression. min and max mean the lower and upper bounds
of constraints, respectively.
The penalty function can be defined as
X
nn X
nm
f cost ðfXgÞ ¼ ð1 þ E1 :vÞE2 W ðfXgÞ, v ¼ vid þ viσ þ viλ ð8:3Þ
i¼1 i¼1
where v is the constraint violations function, υdi , υσi , and υλi are constraint violations
for displacement, stress, and slenderness ratio, respectively, E1 and E2 are penalty
function exponents which were selected considering the exploration and exploita-
tion rate of the search space. Here, E1 is set to unity; E2 is selected in a way that it
decreases the penalties and reduces the cross-sectional areas. Thus, in the first steps
of the search process, E2 is set to 1.5 and it linearly increases to 3 [10].
The allowable tensile and compressive stresses are used according to the AISC-
ASD code [9], as follows:
σþ
i ¼ 0:6 Fy f or σ i 0 ð8:4Þ
σ
i f or σ i < 0
where σ
i is calculated according to the slenderness ratio:
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8.3 CBO and ECBO Algorithms 151
8
>
> λ2 5 3λ λ3
> 1 i 2 Fy = þ i i 3 for λi < Cc
<
2Cc 3 8Cc 8Cc
σ
i ¼ ð8:5Þ
>
> 12π 2
E
>
: for λi Cc
23λ2i
where K is the effective length factor for the members and equal to 1 for all truss
members. Li and ri are the length and minimum radius of gyration for the member i,
respectively.
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152 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Fig. 8.1 Collision between two bodies: (a) before collision, (b) during collision, and (c) after
collision
Fig. 8.2 The sorted CBs in an ascending order and the mating process for the collision
where fit(i) represents the objective function value of the ith CB and n is the number
of colliding bodies. After sorting colliding bodies according to their objective
function values in an increasing order, two equal groups are created: (i) stationary
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8.3 CBO and ECBO Algorithms 153
group and (ii) moving group (Fig. 8.2). Moving objects collide with stationary
objects to improve their positions and push stationary objects toward better posi-
tions. The velocities of the stationary and moving bodies before collision (vi) are
calculated by
n
vi ¼ 0, i ¼ 1, . . . , ð8:8Þ
2
n n
vi ¼ xin2 xi , i ¼ þ 1, þ 2 . . . , n ð8:9Þ
2 2
where xi is the position vector of the ith CB. The velocity of stationary and moving
CBs after the collision (v0i ) is evaluated by
0
miþn2 þ εmiþn2 viþn2 n
vi ¼ i ¼ 1, 2, . . . , ð8:10Þ
mi þ miþn2 2
0
mi εmin2 vi n n
vi ¼ i ¼ þ 1, þ 2, . . . , n ð8:11Þ
mi þ mi2 n 2 2
iter
ε¼1 ð8:12Þ
itermax
where ε is the coefficient of restitution (COR) and iter and itermax are the current
iteration number and the total number of iterations for optimization process,
respectively. New positions of each group are stated by the following formulas:
0 n
xinew ¼ xi þ rand∘vi , i ¼ 1, 2, . . . , ð8:13Þ
2
0 n
xinew ¼ xin2 þ rand∘vi , i ¼ þ 1, . . . , n ð8:14Þ
2
where xinew , xi and v0i are the new position, previous position, and the velocity after
the collision of the ith CB, respectively. rand is a random vector uniformly
distributed in the range of [1,1] and the sign “∘” denotes an element-by-element
multiplication.
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154 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
8.3.2.1 Initialization
where x0i is the initial solution vector of the ith CB. xmin and xmax are the minimum
and the maximum allowable limits vectors, respectively, and random is a random
vector with each component being in the interval [0,1].
8.3.2.2 Search
Step 1: The value of the mass for each CB is calculated by Eq. (8.7).
Step 2: Colliding Memory (CM) is considered to save some historically best CB
vectors and their related mass and objective function values. The size of the CM
is taken as n/10 (n is the population size) in this study. At each iteration, solution
vectors that are saved in the CM are added to the population and the same
number of the current worst CBs are deleted.
Step 3: CBs are sorted according to their objective function values in an increasing
order. To select the pairs of CBs for collision, they are divided into two equal
groups: (i) stationary group and (ii) moving group.
Step 4: The velocities of stationary and moving bodies before collision are evalu-
ated by Eqs. (8.8) and (8.9), respectively.
Step 5: The velocities of stationary and moving bodies after collision are calculated
by Eqs. (8.10) and (8.11), respectively.
Step 6: The new location of each CB is evaluated by Eqs. (8.13) or (8.14).
Step 7: A parameter like Pro within (0, 1) is introduced which specifies whether a
component of each CB must be changed or not. For each CB Pro is compared
with rni (i ¼ 1, 2, . . ., n) which is a random number uniformly distributed within
(0, 1). If rni < Pro, one dimension of ith CB is selected randomly and its value is
regenerated by
xij ¼ xj, min þ random: xj, max xj, min ð8:16Þ
where xij is the jth variable of the ith CB. xj,min and xj,max are the lower and upper
bounds of the jth variable. In this chapter, the value of Pro is set to 0.3.
Step 1: After the predefined maximum evaluation number, the optimization process
is terminated [11].
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8.4 Numerical Examples 155
In this section, two kinds of double-layer barrel vaults are optimized by CBO and
ECBO algorithms and the results are compared with the engineering design which
was found by SAP2000 to show the efficiency of these algorithms. SAP2000
software has a toolbox for the auto and fully stressed design according to the related
provisions. Auto select section lists are lists of previously defined steel sections
(including cold-formed steel). When an auto select section list is assigned to a
frame member, the program can automatically select the most economical, ade-
quate section from the auto select section list when designing the member. The first
example is a 384-bar double-layer barrel vault, which was optimized by Kaveh
et al. [13] using continuous variables under two types of loadings. The second one is
a 910-bar double-layer braced barrel vault introduced as a new type. Two problems
are solved utilizing discrete variables for the purpose of practical design. All
connections are assumed as ball-jointed, and top-layer joints are subjected to
concentrated vertical loads. Stress and slenderness constraints [Eqs. (8.4), (8.5),
and (8.6)] are according to AISC-ASD provisions, and displacement limitations of
0.1969 in (5 mm) are imposed on all nodes in x-, y-, and z-directions. The modulus
of elasticity is considered as 30,450 ksi (210,000 MPa), and the yield stress of steel
is taken as 58 ksi (400 MPa).
In CBO and ECBO, the population of n ¼ 30 CBs is utilized, and the size of
colliding memory is considered as n/10 that is taken as 3 for ECBO. The predefined
maximum evaluation number is considered as 30,000 analyses for all examples.
Because of the stochastic nature of the algorithms, each example is solved 5 times
independently. In all problems, CBs are allowed to select discrete values from the
permissible list of cross sections (real numbers are rounded to the nearest integer in
each iteration). The algorithms are coded in MATLAB, and the structures are
analyzed using the direct stiffness method. The computational time is measured
in terms of CPU time of a PC with the processor of Intel® Core™ i7-3612 QM @
2.1 GHz equipped with 6 GBs of RAM.
Similar to the flat double-layer grids, double-layer barrel vaults consist of a top and
bottom layer connected to each other by bracing members. The top/bottom layers
are also called the “chord members.” All the flat double-layer configurations can
also be used for doublelayer braced barrel vaults. The 384-bar double-layer barrel
vault is the first example; this structure consists of two rectangular nets, and for
making it stable, angles of the bottom nets are put into the center of one of the above
nets, and these are connected through diametrical elements as shown in Fig. 8.3a.
This example is subjected to two types of loadings. Case 1 is a symmetric loading
condition where the vertical concentrated loads of 20 kips (88.964 kN) are
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156 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Fig. 8.3 Schematic of the 384-bar double-layer barrel vault: (a) 3D view and (b) element
grouping (plan view)
applied on free joints (nonsupport joints) of top layer. In Case 2, which is asym-
metric, the concentrated loads of 10 kips (44.482 kN) are applied at the right-
hand half and at the left-hand half of the structure the loads of 6 kips
(26.689 kN) are applied on nonsupport top layer joints, respectively. All members
of this double-layer barrel vault are categorized into 31 groups, as shown in
Fig. 8.3b, and the supports are considered at the two external edges of the top
layer of the barrel vault.
Tables 8.2 and 8.3 show the best design vectors and the corresponding weights
for the two methods, for the Case 1 and Case 2 loading conditions, respectively. In
Case 1 (Symmetric loading condition), ECBO could find the weight which is 2.2 %
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8.4 Numerical Examples 157
Table 8.2 Optimal design of the 384-bar double-layer barrel vault for Case 1
Optimum section (designations)
Group number Engineering design CBO ECBO
1 ST 1¼ ST 1¼ ST 1¼
2 EST 2 EST 2 ST 2½
3 EST 2 EST 3 EST 2
4 ST 1¼ ST 1¼ ST 1¼
5 EST 4 DEST 2 DEST 2½
6 DEST 8 EST 1½ ST 1¼
7 ST 12 EST 10 EST 12
8 EST 8 DEST 5 DEST 5
9 ST 10 DEST 6 ST 10
10 EST 10 EST 10 ST 12
11 ST 8 DEST 5 ST 8
12 EST 8 ST 12 ST 12
13 EST 5 EST 5 DEST 4
14 ST 8 ST 5 ST 6
15 ST 3½ ST 3½ ST 3
16 ST 6 DEST 2½ DEST 2½
17 ST 8 EST 5 EST 5
18 EST 1½ EST 2 ST 2
19 ST 1¼ ST 1¼ ST 1¼
20 EST 2 ST 1¼ ST 2
21 EST 2 EST 2 EST 1¼
22 EST 2 ST 1¼ ST 1¼
23 EST 2 EST 2 ST 2
24 ST 4 EST 3 ST 4
25 ST 2½ EST 2 EST 2½
26 ST 3 EST 2 ST 2½
27 DEST 2½ DEST 2 ST 3½
28 ST 2½ EST 2 EST 2
29 ST 2½ ST 2½ ST 2
30 EST 2 EST 2 ST 2½
31 EST 2 EST 2 EST 2
Demand/capacity ratio limit 0.999 – –
Max stress ratio 0.559 0.7649 0.8773
Max displacement ratio 0.9997 0.9994 0.9999
Best weight (kg) 32,259.90 29,057.93 28,415.20
Mean weight (kg) – 33,465.09 29,900.15
Computation time (s) – 296 291
lighter than CBO and 11.9 % lighter than Engineering design which was found by
SAP2000. In Case 2 (Asymmetric loading condition), this percentage was equal to
10.7 % and 19.7 % better than CBO and Engineering design, respectively. It is also
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158 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Table 8.3 Optimal design of the 384-bar double-layer barrel vault for Case 2
Optimum section (designations)
Group number Engineering design CBO ECBO
1 ST 1¼ ST 1¼ ST 1¼
2 EST 2 EST 2 ST 2
3 ST 1¼ ST 1¼ ST 1¼
4 DEST 2 ST 3½ ST 3½
5 EST 2 EST 2 EST 2
6 EST 2 EST 2 ST 2
7 EST 5 EST 5 EST 6
8 EST 5 ST 8 DEST 4
9 EST 5 DEST 3 EST 3
10 ST 5 ST 3½ ST 3
11 ST 5 ST 3½ ST 3
12 ST 5 EST 3½ ST 4
13 DEST 2 EST 1½ EST 1½
14 ST 2½ ST 3½ ST 2½
15 EST 3 ST 4 ST 4
16 DEST 2½ EST 3½ ST 5
17 ST 5 ST 4 EST 3½
18 EST 1½ ST 1½ ST 1½
19 EST 2 EST 2 ST 2
20 EST 2 EST 2 ST 2
21 EST 2 EST 2 ST 2
22 ST 2½ EST 2 ST 2
23 ST 1½ EST 2 ST 1¼
24 EST 1¼ EST 1 ST 1
25 EST 2 EST 2 ST 1½
26 EST 1½ ST 1½ ST 1½
27 EST 2 EST 2½ ST 2
28 EST 2 EST 2 ST 2
29 EST 2½ EST 2 ST 2
30 DEST 2 ST 2½ ST 3
31 ST 2½ EST 2 ST 2
Demand/capacity ratio limit 0.999 – –
Max stress ratio 0.888 0.7176 0.9372
Max displacement ratio 0.9962 0.9991 0.9996
Best weight (kg) 16,617.81 14,940.13 13,345.92
Mean weight (kg) – 18,602.01 15,856.61
Computation time (s) – 301 299
worthwhile to mention that CBO results were 9.9 % and 10.1 % better than Engi-
neering design for Case 1 and Case 2 loading condition, respectively. It can be
observed that ECBO has better performance than CBO without increasing the
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8.4 Numerical Examples 159
Fig 8.4 Convergence curves for the 384-bar double-layer barrel vault (Case 1)
Fig 8.5 Convergence curves for the 384-bar double-layer barrel vault (Case 2)
computational cost. For graphical comparison of the algorithms, Figs. 8.4 and 8.5
illustrate the convergence curves for the Case 1 and Case 2 loading conditions by
the proposed methods, respectively.
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160 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Braced barrel vaults consist of developable surfaces generated by the repetitive use
of a curve known as “directrix” over a generator straight line. The directrix may be
a circular arc, an ellipse, a catenary, a parabola, or a cycloid. Most of braced barrel
vaults built in practice are part of a right circular cylinder, which may be either
supported by columns or simply springing from the ground surface. The semicir-
cular barrel vaults have the clear advantage of facilitating water drainage and
providing strong architectural form recognition. Under loads, braced barrel vaults
may behave in two different modes: arch and beam, depending mainly on the
location of supports. The braced barrel vault behaves as an arch when supported
along the sides. The braced barrel vault behaves in the beam mode when it is
supported at its ends. In this case, the longitudinal compression forces occur near
the crown and longitudinal tensile forces toward the free edge. If the braced barrel
vault is supported at the four corners, it behaves as a combined beam and arch under
loads. In this case, it acts as a series of arches in cross-section direction and as a
beam longitudinally.
In this section, one type of braced barrel vault which contains 266 nodes and
910 members is introduced as the last example. The structural members are divided
into 30 groups as shown in Fig. 8.6a, and the other related details are shown in
Fig. 8.6b and c.
The uniformity of the distribution of stiffness in the vicinity of the structure is an
important issue for large-scale structures. If part of the structure has elements of low
axial forces and small displacements (low cross sections), and another part contains
elements of high cross sections, then the uniformity of the distribution of the
stiffness will not be achieved. For this reason, the element grouping is selected
according to two symmetry lines of the configuration leading to uniform distribu-
tion of stiffness for the entire structure. The loading conditions consist of the
following:
1. At the nodes of central arc, a downward concentrated load of 15 kips
(66.72 kN).
2. At the nodes of the arcs adjacent to the central arc, a downward concentrated
load of 10 kips (44.48 kN).
3. At the nodes of arcs adjacent to the external arcs, a downward concentrated load
of 5 kips (22.24 kN).
4. At the nodes of external arcs, a downward concentrated load of 2 kips
(8.90 kN).
All external and internal side nodes are simply supported, and for this reason,
this double-layer braced barrel vault behaves as an arch. Table 8.4 lists the optimal
values of 30 variables obtained by ECBO and CBO. The result of ECBO method is
lighter than the result found by CBO. The optimum design for CBO and ECBO has
the weights of 18,636 kg and 18,615 kg, respectively, and all optimum designs
found by the algorithms satisfy the design constraints. The CBO and ECBO weights
are 1258.77 kg (6.3 %) and 1279.12 kg (6.4 %) lighter than Engineering design,
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8.4 Numerical Examples 161
Fig 8.6 Schematic of the 910-bar double-layer braced barrel vault: (a) element grouping in 3D
view, (b) front view, and (c) plan view
respectively. Convergence history of the present algorithms for the best optimum
designs is depicted in Fig. 8.7.
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162 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
Table 8.4 Optimal design of the 910-bar double-layer braced barrel vault
Optimum section (designations)
Group number Engineering design CBO ECBO
1 DEST 2 EST 2 ST 3
2 DEST 5 ST 10 DEST 6
3 ST 8 ST 10 ST 10
4 ST 8 DEST 2½ ST 8
5 DEST 2 ST 3½ ST 2
6 EST ¾ EST ¾ ST 1½
7 ST 4 EST 3 EST 1½
8 ST 8 DEST 3 EST 3
9 ST 10 DEST 5 DEST 4
10 ST 12 DEST 6 EST 12
11 ST 1 ST 1¼ ST 1
12 ST 1 ST 1¼ ST 1
13 EST 1 ST 1¼ ST 1¼
14 ST 1 ST 1 ST 1
15 EST 2 ST 1 ST 1
16 ST 1 ST 3 EST 2
17 EST 1½ EST 1½ EST 2
18 EST 1½ EST 1 EST 1½
19 ST 1¼ EST 2 EST 3
20 EST 2 ST 2½ EST 2
21 EST 3½ ST 2½ EST 2
22 DEST 2½ ST 2½ EST 2
23 ST 5 EST 4 DEST 3
24 EST 5 ST 8 EST 5
25 ST ¾ ST ¾ ST ¾
26 ST ¾ ST ¾ ST ¾
27 ST ¾ ST½ ST ¾
28 ST ¾ ST ¾ EST ¾
29 EST 2 ST ¾ ST ¾
30 EST 1 EST 2 EST 1½
Demand/capacity ratio limit 0.999 – –
Max stress ratio 0.95 0.9767 0.9818
Max displacement ratio 0.9993 0.9990 0.9978
Best weight (kg) 19,894.44 18,635.67 18,615.32
Mean weight (kg) – 23,806.75 22,442.64
Computation time (s) – 975 926
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8.5 Concluding Remarks 163
Fig. 8.7 Convergence curves for the 910-bar double-layer braced barrel vault
This chapter utilizes two newly developed, simple, and efficient metaheuristic
algorithms for discrete optimization of double-layer barrel vaults. The CBO has
simple structure and depends on no internal parameter and does not use memory for
saving the best-so-far solutions. In order to improve the exploration capabilities of
the CBO and to prevent premature convergence, a stochastic approach is employed
in ECBO that changes some components of CBs randomly. Colliding Memory is
also utilized to save a number of the so-far-best solutions to reduce the computa-
tional cost. In order to indicate the similarities and differences between the char-
acteristics of the CBO and ECBO algorithms, two types of double-layer barrel
vaults are examined. Structures are designed in accordance with AISC-ASD spec-
ifications and displacement constraints. In both examples, the discrete variables are
assigned to each group for the purpose of practical design and selected from
available steel pipe section table. ECBO has better performance in all cases than
CBO because of the reliability of search, solution accuracy, and speed of conver-
gence. It can be also stated that both CBO and ECBO have better efficiency in
finding results than SAP2000 in all cases. It is also worthwhile to mention that all
designs are governed by displacements because of large vertical displacements at
the apex of these structures. Furthermore, the results show that CBO and ECBO are
robust optimization tools for optimum practical design of large-scale structures like
double-layer barrel vaults.
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164 8 Optimal Design of Double-Layer Barrel Vaults Using CBO and ECBO Algorithms
References
1. Kaveh A, Moradveisi M (2016) Optimal design of double-layer barrel vaults using CBO and
ECBO algorithms. Iran J Sci Technol Trans Civil Eng. doi:10.1007/s40996-016-0021-4
2. Makowski ZS (2006) Analysis, design and construction of braced barrel vaults. Taylor &
Francis e-Library, Hoboken, NJ
3. Kaveh A, Mirzaei B, Jafarvand A (2013) Optimal design of single-layer barrel vault frames
using improved magnetic charged system search. Int J Optim Civil Eng 3(3):575–600
4. Kaveh A, Mirzaei B, Jafarvand A (2014) Shape-size optimization of single-layer barrel vaults
using improved magnetic charged system search. Int J Civil Eng IUST 12(4):447–465
5. Kaveh A (2014) Advances in metaheuristic algorithms for optimal design of structures.
Springer, Switzerland
6. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel metaheuristic method.
Comput Struct 139:18–27
7. Kaveh A, Ilchi Ghazaan M (2014) Computer codes for colliding bodies optimization and its
enhanced version. Int J Optim Civil Eng 4:321–332
8. American Institute of Steel Construction (AISC) (1994) Manual of steel construction load
resistance factor design, 2nd edn. AISC, Chicago, IL
9. American Institute of Steel Construction (AISC) (1989) Manual of steel construction allow-
able stress design (ASD-AISC), 9th edn. AISC, Chicago, IL
10. Kaveh A, Farahmand Azar B, Talatahari S (2008) Ant colony optimization for design of space
trusses. Int J Space Struct 23:167–181
11. Kaveh A, Ilchi Ghazaan M (2015) A comparative study of CBO and ECBO for optimal design
of skeletal structures. Comput Struct 153:137–147
12. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies algorithm for truss optimization
with dynamic constraints. J Comput Civil Eng. doi:10.1061/(ASCE)CP.1943-5487.0000445,
04014104
13. Kaveh A, Mirzaei B, Jafarvand A (2014) Optimal design of double layer barrel vaults using
improved magnetic charged system search. Asian J Civil Eng 15(1):135–154
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Chapter 9
Optimum Design of Steel Floor Systems
Using ECBO
9.1 Introduction
Decks, interior beams, edge beams, and girders are parts of a steel floor system. If
the deck is optimized without considering beam optimization, finding the best result
is simple. However, a deck with a higher cost may increase the composite action of
the beams and decrease the beam cost, thus reducing the total expense. Also, a
different number of floor divisions can improve the total floor cost. Increasing beam
capacity by using castellated beams is another efficient cost-saving method. In this
study, floor optimization is performed and these three issues are discussed. Floor
division number and deck sections are some of the variables. Also, for each beam,
profile section of the beam, beam-cutting depth, cutting angle, spacing between
holes, and number of filled holes at the ends of castellated beams are other vari-
ables. Constraints include the application of stress, stability, deflection, and vibra-
tion limitations according to the load and resistance factor (LRFD) design. The
objective function is the total cost of the floor consisting of the steel profile, cutting
and welding, concrete, steel deck, shear stud, and construction costs. Optimization
is performed by enhanced colliding bodies optimization (ECBO). Results show that
using castellated beams, selecting a deck with a higher price and considering the
different number of floor divisions can decrease the total cost of the floor (Kaveh
and Ghafari [1]).
Many researchers have tried to optimize simple, composite, and castellated
beams. Morton and Webber [2] used a relatively straightforward exhaustive search
method to optimize composite beams. Klanšek and Kravanja [3] utilized the
nonlinear programming (NLP) approach to optimize composite beams according
to Euro-code 4 and conditions of both ultimate and serviceability limit states.
Senouci and Al-Ansari [4] optimized composite beams by genetic algorithms
according to AISC-LRFD. They also tried to find the effect of span and loading
on the optimum result by a parametric study.
Cost optimization of floor systems is studied first by Adeli and Kim [5]. They
utilized neural networks and mixed integer nonlinear programming according to the
LRFD criteria. They also employed floating-point genetic algorithms to find the
best results. Platt [6] used the evolver (genetic algorithm solving program) to
parametric optimization of the floor. She considers the combination of configura-
tion, size, topology, and spacing of truss girders and beams. Kaveh and Abadi [7]
used an improved harmony search (HS) algorithm. They optimized a composite
floor system consisting of reinforced concrete slab and steel I-beams according to
AISC-LRFD rules. Poitras et al. [8] considered a complete floor system and utilized
particle swarm optimization (PSO) for optimization. They found that composite
action can be as economical as non-composite action depending on some condi-
tions, and they used formed steel deck instead of normal concrete deck. Kaveh and
Ahangaran [9] employed the social harmony search and found this new variant of
HS to be better than other variants of it. Kaveh and Massoudi [10] optimized floors
by ant colony optimization (ACO).
The main objective of the present chapter is to optimize the cost of the steel floor
elements and to find the effect of the number of floor divisions, concrete thickness,
and using castellated beams. This chapter is organized as follows: In Sect. 9.2, the
design of structural elements of floor is introduced. Section 9.3 defines the optimi-
zation problem and identifies the variables, the constraints, and the objective
function. The optimization algorithm is discussed in Sect. 9.4. Some numerical
examples are introduced in Sect. 9.5. Finally, conclusions are extracted in Sect. 9.6.
Structural elements are designed according to AISC-LRFD 10. Thus, the load
combination W for stress and stability check is (ASCE [11])
W ¼ 1:2 DL þ 1:6 LL
where DL is the dead load and LL is the live load, and the load combination for
serviceability criteria (deflection and vibration) is
W def ¼ DL þ LL
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9.2 Structural Floor Design 167
Deck span is the distance between two beams (B), and deck width is taken as
1 meter for the design. In this study, composite steel deck is used so that its section
shape can guarantee the composite action roll formed steel decks and concrete. Also
the shrinkage and temperature effects of the concrete are controlled by rebar. Due to
the complex effect of roll formed steel decks, the partial composite action, and the
wide variety of the produced sections, the specifications provided by the manufac-
turers should be used for determining their capacity.
Castellated beams are produced by cutting rolled profile beam in special shape and
welding them together in order to increase moment of inertia and moment capacity.
Hexagonal cutting shape is one of the most popular cutting methods. But it is
necessary to avoid keen corners because of stress concentration effects. Web
openings of these beams produce some secondary effects, which can be controlled
by filling end holes.
Composite beams are produced by composite interaction between concrete and
steel. This composite action can help to increase the moment capacity of the beams.
For designing this type of beams, first the effective width of the concrete slab
should be calculated for interior beams, edge beams, and girders according to span
and beam spacing (AISC [12]). Second, for the composite section, the center line
must be calculated. For interior and edge beams, deck ribs are perpendicular to the
beam axis, and top concrete (Fig. 9.1) must be considered only. However, for
girders, the deck ribs are parallel to the beam axis and the entire concrete can be
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168 9 Optimum Design of Steel Floor Systems Using ECBO
considered (AISC [12]). In this study, the center line, the moment of inertia, and the
moment capacity of the composite section are determined by the superposition of
the elastic stresses. For some stresses, stability, deflection, and vibration criteria
must be checked as follows.
In this study, the unbraced length ratio of all beams is considered as zero. This is
because the top flange of the beam is controlled by concrete slab.
The ultimate moment calculated for load combinations must be smaller than the
nominal moment (AISC [12]):
where Mn is the nominal moment capacity of the beam, Mn-con is the nominal
moment capacity (concrete limit), Mn-st is the nominal moment capacity (steel
limit), Znet-com-bot is the plastic modulus at the bottom of composite net section,
Znet-com-top is the plastic modulus at the top of composite net section, φb is the
bending reduction factor, Fc is the compressive strength of the concrete, and Fy is
the yield strength of the steel.
Also the Vierendeel effect at unfilled holes produces secondary moment, and
these two moments must satisfy the following equations:
Vu e
mu ¼ ð9:2Þ
4
Mu mu
þ < φb F y ð9:3Þ
Z net-com-bot Z tee
where mu is the secondary shear ultimate moment, Vu is the ultimate shear force, e is
the web post length, Mu is the ultimate moment, Znet-st is the plastic modulus of steel
net section, and Ztee is the plastic modulus of steel tee section. φb for concrete and
steel are considered to be 0.9 (AISC [12]).
For a composite section, steel beams must resist shear forces alone (AISC [12])
as described in the following:
AW ¼ d s tw ð9:4Þ
V u < φv V nw ¼ φv 0:6Fy AW Cv ð9:5Þ
where Aw is the area of the net section web, tw is the thickness of the web, ds is the
internal castellated beam height, Vu is the ultimate shear force, Vnw is the nominal
web shear capacity of net section, φv is the shear reduction factor, and Cv is the web
shear coefficient.
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9.2 Structural Floor Design 169
Also the vertical shear capacity of the tee beams must be controlled by (AISC
[12]):
where Atee is the area of each tee section and Vntee is the nominal web shear
capacity of the tee section.
Horizontal shear between holes in castellated beams must be checked as follows:
Ahe ¼ e tw ð9:8Þ
V u Qcom
Vh ¼ s < φv V np ¼ φv 0:6Fy Ahe Cv ð9:9Þ
I com
where Vh is the horizontal shear at web post; Qcom and Icom are the first and second
moments of inertia of the composite section, respectively; s is the spacing between
the holes (Fig. 9.1); Vnp is the nominal shear capacity of the web post; and φv and
Cv are equal to 1 (AISC [12]).
When steel deck is used in a perpendicular position, Qcom and Icom must be
considered for two conditions, because each choice may produce a greater shear
force and a more critical condition:
(a) Considering the whole thickness of the concrete
(b) Considering the top thickness of the concrete
Horizontal shear may cause web plate buckling in the castellated beam (Kerdal and
Nethercot [13]). According to the Structural Stability Research Council (SSRC),
in-plane stress at the unfilled web must satisfy the following equations:
Lb ¼ 2dh
tw
r T ¼ pffiffiffiffiffi
12 2
M1 M1
Cb ¼ 1:75 þ 1:05 þ 0:3 < 2:3
M2 M2
2π 2 Es ð9:10Þ
Cc ¼
Fy 2 2 3
Lb
3 V h tan θ 6 rT 7
f rb ¼ < φb Frb ¼ 41 5 φb Fy
4 tw θ 2 e 2Cc 2 Cb
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170 9 Optimum Design of Steel Floor Systems Using ECBO
where θ, e, and dh are the cutting angle, hole pure distance, and cutting depth of
castellated beam, respectively (Fig. 9.1), tw is the thickness of the web, M1 and M2
are the moments at each beam end, Es is the modulus of elasticity of the steel, and
φb is equal to 0.9 similar to the moment equation.
5W d1 LT 4 5W d2 LT 4
def b ¼ þ ð9:11Þ
384Es I n 384Es I def
where Wd1 and Wd2 are the pre-composite and post-composite loads, respectively,
LT is the total beam length, and Idef and In are the effective moment of inertia for
deflection of composite beam and steel net section moment of inertia, respectively.
Concrete weight must be resisted by steel section only (pre-composite level),
and other dead and live loads must be sustained by composite section (post-
composite level).
Deflection of the girders is related to the number floor divisions (beam spacing)
and the number of interior beams.
Unlike the standard composite beam, the shear deflection of the composite beam
with web opening is significant. Thus researchers have developed experimental-
based equation for calculating the shear deflection (defs) as follows (Benitez et al.
[14]):
1 HEW I com HEW 3 HEW 2
def s ¼ def b 1 þ 1 3 4
5 LT I comg LT LT
!!
HEW
6 þ 12 ð9:12Þ
LT
where Icom and Icomg are the net and gross composite section moments of inertia,
respectively. This equation is based on rectangular shape holes, and the hexagonal
shapes must be considered as rectangular shapes with effective width:
and defs identifies the effect of one hole. For web opening with a width to height
ratio lower than 2, maximum deflection of the beam is independent of the location
of the holes. Thus, the total shear deflection can be obtained from the number of
unfilled holes (Nuh) times the defs, and the total beam deflection is calculated as
follows:
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9.2 Structural Floor Design 171
Also for considering the effect of differential shrinkage and creep on a composite
steel–concrete structure, the effective width (or concrete modulus of elasticity) can
be divided by 3 (Roll [15]).
Also, the allowable deflection (defall) under the live and dead loads is specified
by AISC [12] as
LT
def < def all ¼ ð9:14Þ
240
A portion of the live load (between 10 and 25 %) that is used for calculating
deflection is utilized for calculating vibration (defvib) (Murray et al. [16]). Com-
bining the effect of the interior beam deflection (defint), the girder beam deflection
(defgir), and column deflection (defcol) for calculating frequency is considered as
follows (Naeim [17]):
In order to take into account the difference between the frequency of a simply
supported beam with distributed
mass and concentrated mass at mid-span, the
deflection is divided by 1.3 π4 (Murray et al. [16]).
Because of the small compression deflection of the column, defcol is considered
as zero. Also, 0.2 times of the live load is used in calculating the deflection.
For considering greater stiffness of concrete on the metal deck under dynamic
loading compared to the static loading, it is assumed that the modulus of elasticity
for the concrete is 1.35 times that of the normal concrete. The effect of differential
shrinkage and creep on a composite steel–concrete structure is not considered for
vibration calculations.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffi
u W rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi
1 Stiffness 1 u
t def vib g 1 g
f ¼ ¼ W ¼ ð9:16Þ
2π Mass 2π g
def vib 2π def vib
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172 9 Optimum Design of Steel Floor Systems Using ECBO
1 1 1 1
¼ þ þ ð9:17Þ
f t f int f gir f col
where fint, fint, and fint are the interior, girder, and column frequencies, respectively.
Due to the large axial stiffness of the column in comparison to the bending
stiffness of beams, column frequency is considered infinity.
The maximum initial amplitude (inch) of the beam (Ao) is determined as (Naeim
[17])
!
0:6ðLT 0:393Þ3
Aot ¼ ðDLFÞmax ð9:18Þ
48 Es 14:22 103 I def 0:3934
where Sb is the beam spacing. (DLF)max values for various natural frequencies are
presented in design practice to prevent floor vibrations (Naeim [17]). Effective
concrete height (hceff) is not equal to the concrete height in the steel deck floor.
Required damping ratio (Dreq) for specified amplitude and frequency must be lower
than the allowable damping ratio (Dall), and it is determined as (Naeim [17])
For a desired composite action between steel and concrete, shear studs are required.
The shear capacity of these elements must be larger than the maximum shear forces
that composite beam will experience. Steel-headed stud anchor is considered in this
chapter. Its diameter is considered as 19 mm and 1, 2, or 3 studs can be installed at
each rib.
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9.3 Problem Definition 173
pffiffiffiffiffiffiffiffiffiffi
Qu ¼ min 0:85Fc be hc , As Fy < N c φv Qn ¼ N c φv 0:5Asa Fc Ec
Rg Rp Asa Fuss ð9:23Þ
where Fc and Ec are the compression strength and modulus of elasticity of concrete,
respectively; be and hc are the effective width and height of concrete, respectively;
As and Asa are the steel section area and steel-headed shear stud area, respectively;
Fuss is the ultimate stress of shear stud; and Rg and Rp are the group and position
effect factor for shear stud, respectively. Considering linear shear diagram, Nc is
half of the total number of shear stud and φv is equal to 0.75 (AISC [12]).
The cost for each beam is considered as the sum of the profile steel beam cost,
welding procedure cost, cutting procedure cost, and shear stud cost. The cost for
steel deck is the sum of the steel deck concrete cost, steel deck steel plate cost, and
steel deck application cost. Initial cost is the sum of the beam costs and steel
deck cost.
Each sub-cost is determined by multiplying the corresponding weight, length,
volume, or area by appropriate coefficients. Cost of filling end holes by plates is
considered by the cost of the added weights, cutting, and welding to the total cost.
9.3.2 Variables
In this chapter, five variables are used for optimal design of each beam, consisting
of the profile section, cutting depth (dh), cutting angle (α), hole spacings (s), and
number of filled end holes of the castellated beams. The number of beams at
floor width and concrete thickness are two other variables that are changed. The
minimum and maximum magnitudes of the variables must be known for avoiding
unacceptable results and for fast convergence to the global optimum. Profile
section is the sequence number of the hot rolled steel profiles. Cutting angle is
limited between 40 and 64 . Other limits on the variables are presented as the
constraints.
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174 9 Optimum Design of Steel Floor Systems Using ECBO
9.3.3 Constraints
Castellated beam application constraints (g1 to g5) and steel beam design con-
straints (g6 to g14) are considered as follows:
3
g1 ¼ dh ðHs 2tf Þ ð9:24Þ
8
g2 ¼ ðH s 2tf Þ 10ðd t tf Þ ð9:25Þ
2
g3 ¼ d h cot ðαÞ e ð9:26Þ
3
g4 ¼ e 2d h cot ðαÞ ð9:27Þ
g5 ¼ 2dh cot ðαÞ þ e 2d h ð9:28Þ
g6 ¼ M u φ b M n ð9:29Þ
Mu mu
g7 ¼ þ φb F y ð9:30Þ
Znet-com-bot Ztee-com
g8 ¼ V u φv V nw ð9:31Þ
Vu
g9 ¼ φv V ntee ð9:32Þ
2
g10 ¼ V h φv V np ð9:33Þ
g11 ¼ f rb φb Frb ð9:34Þ
g12 ¼ def def all ð9:35Þ
g13 ¼ Dreq Dall ð9:36Þ
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9.4 Optimization Algorithm 175
avoids infeasible areas. In this study, penalty function is expressed as the function
of positive (unacceptable) values of the constraint functions:
where Costfin and Costini are the final and initial costs, respectively. The value of
10 is chosen by the experience for the current problem and it can be changed for
other problems.
Interior beam optimization, edge beam optimization, girder optimization, and deck
optimization are four suboptimizations of this problem. Each of the first three
problems has five variables according to the explanation given in the previous
section. Deck optimization has one variable and the number of floor division is
another variable. Thus, there are 17 optimization variables for this problem. Opti-
mizing these variables simultaneously decreases the convergence rate. In order to
solve this problem, and to observe the conditions around the optimum result, the
following approach is adopted.
If the deck is optimized without considering beam optimization, finding the best result
is simple. But other decks with higher costs can increase composite action of the
beams and decrease the beam cost, hence reducing the total cost. Thus, after finding
the best deck independently (by sorting deck choices from lower to highest cost and
selecting the first acceptable choice), some other near acceptable results are consid-
ered, and optimum result of other parts of the floor is calculated for the entire system.
The range for the number of divisions of the floor is limited for different
examples. To observe the impact of increasing the number of division, different
values are considered and the results of optimization are obtained.
In order to optimize each beam, the following metaheuristic algorithm is used:
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176 9 Optimum Design of Steel Floor Systems Using ECBO
them. The algorithm produces the next generation from the initial population in order
to increase the chance of find the best result. So increasing the number of population
and iteration number can increase the chance of finding the optimum result.
Colliding bodies optimization is one of the recently developed metaheuristic
algorithms. The efficiency of this algorithm for structural optimization is validated
by researchers (Kaveh and Mahdavi [18]). The CBO is simple in concept and
depends on no internal parameter.
In this technique, one object collides with other objects and they move toward a
minimum energy level. Each colliding body (CB) has a specified mass (mk) related
to the fitness function as
1
fitðkÞ
mk ¼ Xn ð9:41Þ
1
, k ¼ 1, 2, . . . , n
i¼1
fit ðiÞ
where fit and n are the fitness function and the number of CBs, respectively. In order
to select pairs of objects for the collision, CBs are sorted according to the magni-
tudes of their mass in a decreasing order, and are divided into two equal groups: a
stationary group and a moving group. Moving objects collide with stationary objects
to improve their positions and push stationary objects toward better positions by
changing their velocity. Initial velocity of the moving objects (v1) is defined as a
distance between their positions and destination of the stationary object. Initial
velocity of stationary objects is considered as zero. Next, velocity of stationary
(vsta) and moving (vmov) groups is calculated as follows:
where m1, m2, v1, and v2 are the mass and velocity of each pair of moving and
stationary objects. Also, ε is defined as follows:
iter
ε¼1 ð9:44Þ
iter max
where iter and itermax are the current iteration number and maximum iteration
number, respectively. Next, position of each CB is its last position plus a random
ratio of velocity.
In order to improve the CBO to get faster and more reliable solutions, enhanced
colliding bodies optimization (ECBO) has been developed which uses a memory to
save a number of historically best CBs and also utilizes a mechanism to escape from
local optima (Kaveh and Ilchi Ghazaan [19]). Utilizing this improvement requires
to identify the colliding memory size (CMS) and the random parameter (RP).
Flowchart of the analysis and optimization of floor system is shown in Fig. 9.2.
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9.5 Numerical Examples 177
In order to study the effect of parameters on the optimum cost of the floor, two
examples are studied. MATLAB software is used for modeling the optimization
process. This software is also used for the analysis and checking design criteria. The
design results are also double-checked with ETABS software.
In both examples, floor systems with two girders, two edge beams, and some
interior beams are considered as shown in Fig. 9.3, and all connections are assumed
pinned connections.
For algorithm adjustments, the population size and the iteration number are
40 and 60, respectively. Also, CMS and RP are considered to be 4 and 0.3,
respectively.
At the first example, the span and width of the floor system are 10 m and 8 m,
respectively. Interior beams are affected by live and dead area loads. Edge and
girder beams are affected by live and dead uniformly distributed loads (in order to
take the influence of adjacent bay and wall load into account). Girder beam is also
affected by end reaction of interior beam as a point load.
Full composite action is considered, since partially composite action is very
sensitive to construction and installation conditions of shear studs and it has a large
amount of uncertainty.
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178 9 Optimum Design of Steel Floor Systems Using ECBO
W
Girder
Edge beam
Edge beam
Interior beam
Interior beam
Interior beam
L
Girder
Fig. 9.3 Floor system configuration for the floor division number is equal to 4
In order to have a comparison with other reference examples (Poitras et al. [8]),
the steel deck choices were taken from the Canam® steel catalogue as presented in
Fig. 9.4. According to P-2434 (composite type of this catalogue), deck thickness
values are considered as 0.76, 0.91, and 1.21 mm. Slab thickness values are taken as
125, 140, 150, 165, 190, and 200 mm. Maximum span for each combination of deck
and steel thickness is determined and the load resistance for each span is calculated.
It is assumed that each span has adjacent span in the start and end (triple span
condition). Shoring decks are not considered.
The profile sections are chosen by the Canadian Handbook of the Steel Con-
struction, starting from W410 39 and ending with W690 289. The steel yield-
ing stress, steel modulus of elasticity, and concrete compression capacity are
3550 kg/cm2, 2,050,000 kg/cm2, and 200 kg/cm2, respectively.
The values of the cost coefficients are determined by other researchers (Poitras
et al. [8]) and engineering experiences. Cost coefficients are given in Table 9.1.
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9.5 Numerical Examples 179
Poitras et al. [8] did not consider the effect of shrinkage and temperature as
discussed below. In order to compare the results of this study with their results,
shrinkage and temperature effects are not considered in Example 1. They also used
the S16 standard requirements (CSA [20]). Penalty factors in their work were
considered constant and this assumption decreased the convergence rate.
For comparing results with other researchers and presenting effect of castellated
beams, four problem types are assumed and they are defined in Table 9.2. Also final
costs of each type are presented in this table.
Critical constraints (over 80 % demand capacity ratio) are shown in Table 9.3.
Also, detailed results include the section profile of each beam as presented in
Table 9.4.
The results of Example 1 are shown for comparison, and 4 % difference is
observed between the results of Poitras et al. [8] and the checked values. It should
be mentioned that they considered 75 % for composite action and our study
considers full composite action. Thus, the number of shear studs is lower than our
study.
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180 9 Optimum Design of Steel Floor Systems Using ECBO
Table 9.5 Effect of the floor division number and deck section on the total cost (Example 1)
Deck price
Composite beams Composite castellated beams
Floor division number Low Medium High Low Medium High
2 20,197 17,851 17,207 19,762 16,350 17,484
3 14,892 14,170 14,323 14,422 13,969 14,446
4 14,399 14,097 14,639 12,796 13,312 13,842
5 14,208 14,274 14,444 13,781 15,440 14,057
This example is similar to Example 1. Span and width are 6 m and 7 m, respec-
tively. The profile sections are chosen from the IPE steel sections, starting from
IPE140 and ending with IPE600. The steel yielding stress, steel modulus of
elasticity, and concrete compression capacity are 2400 kg/cm2, 2,039,000 kg/cm2,
and 250 kg/cm2, respectively. The effects of shrinkage and temperature are consid-
ered. There is no uniform distributed load on edge beams and girders. In order to
simulate adjacent bay conditions, they also resist two times of typical load of the
exiting bay. Because the same loading was used on the interior and edge beams,
their results are presented together. Other parameters of Example 2 are similar to
those of Example 1.
Critical constraints (over 80 %), detailed results, and costs of the choices are
shown in Table 9.6, Table 9.7, and Table 9.8, respectively. Also, hole spacing for
cutting depths is extracted from detailed results for beams, and the average of these
ratios is calculated and presented in Table 9.9.
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9.6 Concluding Remarks 181
Optimization and parametric studies of steel floor systems with composite and
castellated beams and steel decks are performed in this study. The objective
function is the floor cost where 17 variables and parameters are considered. The
stress, stability, deflection, and vibration criteria are all discussed. Results indicate
that:
1. Using the high-price decks in order to amplify the composite action can improve
the results and decrease the cost between 5 and 10 % in composite beams and
composite castellated beams. It seems that choosing the most expensive deck
does not guarantee the best result. So considering the first three acceptable decks
is a good assumption.
2. Considering different number of divisions can decrease the total cost between
10 and 20 %.
3. Using composite castellated beams improves the results by about 14 % com-
pared to the composite beams.
4. The optimum degree of castellated cutting angle is about 63 .
5. Average ratio of hole spacing to cutting depth is between 2 and 3. This ratio is 3
for commercial castellated beams
The results show that the utilized optimization algorithm, ECBO, performs quite
well, and it has reliable and accurate solution. The fast-converging feature of the
standard CBO is generally preserved in ECBO, whereas the modifications of the
latter algorithm improve the exploration capabilities of the CBO. One can conclude
that ECBO algorithm is competitive with the other available optimization methods.
For an extensive comparative study of ECBO, when applied to different structural
optimization problems, one can refer to Kaveh [21].
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182
number price Section (cm) (d) (cm) hole Section (cm) (d) (cm) hole (cm) (cm)
3 Low IPE500 13.93 62.48 37.19 0 IPE240 7.95 63.8 23.07 0 12.5 0.91
Medium IPE500 13.93 62.48 37.19 0 IPE240 7.95 63.8 23.07 0 12.5 0.91
High IPE450 9.93 63.67 29.49 0 IPE240 5.88 63.7 17.04 0 14 0.91
4 Low IPE360 10.96 63.98 31.05 0 IPE220 8.07 63.8 17.69 0 12.5 0.76
Medium IPE400 8.16 62.69 21.79 0 IPE240 11.7 63.2 23.96 0 14 0.76
High IPE300 17.46 63.78 50.54 3 IPE220 6.82 63.5 15.79 0 12.5 0.91
5 Low IPE330 9.81 63.80 26.81 0 IPE200 8.66 59.9 19.38 0 12.5 0.76
Medium IPE300 8.62 62.88 25.51 0 IPE180 6.37 64 13.66 0 14 0.76
High IPE330 12.61 63.16 30.47 0 IPE200 4.32 59.4 11.1 0 12.5 0.91
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Optimum Design of Steel Floor Systems Using ECBO
References 183
References
1. Kaveh A, Ghafari MH (2016) Optimum design of steel floor system: effect of floor division
number, deck thickness and castellated beams. Struct Eng Mech 59(5):933–950
2. Morton S, Webber J (1994) Optimal design of a composite I-beam. Compos Struct 28
(2):149–168
3. Klanšek U, Kravanja S (2007) Cost optimization of composite I beam floor system. Am J Appl
Sci 5(1):7–17
4. Senouci AB, Al-Ansari MS (2009) Cost optimization of composite beams using genetic
algorithms. Adv Eng Softw 40(11):1112–1118
5. Adeli H, Kim H (2001) Cost optimization of composite floors using neural dynamics model.
Commun Numer Methods Eng 17(11):771–787
6. Platt BS, Mtenga PV (2007) Parametric optimization of steel floor system cost using evolver.
WIT Trans Built Environ 91:119–128
7. Kaveh A, Abadi ASM (2010) Cost optimization of a composite floor system using an improved
harmony search algorithm. J Constr Steel Res 66(5):664–669
8. Poitras G, Lefrançois G, Cormier G (2011) Optimization of steel floor systems using particle
swarm optimization. J Constr Steel Res 67(8):1225–1231
9. Kaveh A, Ahangaran M (2012) Discrete cost optimization of composite floor system using
social harmony search model. Appl Soft Comput 12(1):372–381
10. Kaveh A, Massoudi M (2012) Cost optimization of a composite floor system using ant colony
system. Iran J Sci Technol Trans Civil Eng 36(C2):139–148
11. ASCE (1994) Minimum design loads for buildings and other structures, vol 7. American
Society of Civil Engineers, Chicago, IL
12. AISC (2010) Specification for structural steel buildings (ANSI/AISC 360-10). American
Institute of Steel Construction, Chicago, IL
13. Kerdal D, Nethercot D (1984) Failure modes for castellated beams. J Constr Steel Res 4
(4):295–315
14. Benitez MA, Darwin D, Donahey RC (1998) Deflections of composite beams with web
openings. J Struct Eng ASCE 124(10):1139–1147
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184 9 Optimum Design of Steel Floor Systems Using ECBO
15. Roll F (1971) Effects of differential shrinkage and creep on a composite steel-concrete
structure. ACI Spec Publ 27
16. Murray TM, Allen DE, Ungar EE (2003) Floor vibrations due to human activity. American
Institute of Steel Construction, Chicago, IL
17. Naeim F (1991) Design practice to prevent floor vibrations. In: Steel Tips, Structural Steel
Educational Council, Technical Information and Product Service, Steel Committee of
California
18. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 139:18–27
19. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
20. Csa C (2009) CSA-S16-09: design of steel structures. Canadian Standards Association,
Mississauga, ON
21. Kaveh A (2014) Advances in metaheuristic algorithms for optimal design of structures.
Springer, Switzerland
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Chapter 10
Optimal Design of the Monopole Structures
Using the CBO and ECBO Algorithms
10.1 Introduction
Tubular steel monopole structures are widely used for supporting antennas in
telecommunication industries. This chapter utilizes two recently developed
metaheuristic algorithms, so-called colliding bodies optimization (CBO) and its
enhanced version (ECBO), for size optimization of monopole steel structures. The
optimal design procedure aims to obtain minimum weight of monopole structures
subjected to the TIA-EIA222F specifications. Two numerical examples are exam-
ined to verify the suitability of the design procedure and to demonstrate the
effectiveness and robustness of the CBO and ECBO in creating optimal design for
this problem. The outcomes of the ECBO are also compared to those of the standard
CBO to illustrate the importance of the enhancement of the CBO algorithm [1].
Over the last decade, there has been an increasing use of cellular telephones,
including new smartphones, for voice and data communication, and wireless
Internet access, which has increased the demand for wireless data transmission
bandwidth. As a result, there has been a large increase in the number of monopoles
installed around populated areas to support antennas. Monopoles have become an
important part of our communications infrastructure [2–4]. Therefore, optimal
design of the monopole structures can be an interesting and challenging issue in
the structural engineering research.
The monopole structures can be categorized based on cross-sectional variations
along height into two types: the tapered type and stepped type. In tapered type the
cross section is continuously decreasing from bottom to top of monopole, and in
stepped type the structure is divided into some parts with abrupt changes between
sections [2]. The sections of stepped monopoles can be circular and polygonal in
shape [5]. Figure 10.1 shows the schematic shape of a treble-part-monopole with
circular sections. The main objective of this chapter is to find the optimum size of
sections of the steel circular stepped monopoles. Here, the CBO and ECBO
algorithms are utilized for optimization, where the weight of the monopole is
Fig. 10.1 The circular treble-part-monopole: (a) Three-dimensional view, (b) front view
considered as the objective function. The design method used in this chapter is also
consistent with TIA-EIA222F specifications [6].
Optimization algorithms can be divided into two categories: (1) local optimizers
and (2) global optimizers. Local optimizer algorithms which often utilize the
gradient information or iterative methods to search the solution space near an initial
starting point by local changes, are hard to apply and time-consuming in these
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10.2 Monopole Structure Optimization Problem 187
Find X ¼ ½x1 ; x2 ; x3 ; . . . ; xn
to minimizes MerðXÞ ¼ f ðXÞ f penalty ðXÞ
ð10:1Þ
subjected to gi ðXÞ 0, i ¼ 1, 2, . . . , m
ximin xi ximax
where X is the vector of design variables with n unknowns, gi is the ith constraint
from m inequality constraints, Mer(X) is the merit function, f(X) is the cost function,
fpenalty(X) is the penalty function which results from the violations of the constraints
corresponding to the response of the monopole structures, and also ximin and ximax
are the lower and upper bounds of the design variable vector, respectively.
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188 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
X
m
f penalty ðXÞ ¼ 1 þ γ p max 0, gj ðxÞ ð10:2Þ
i¼1
The most effective parameters for creating the monopole structure geometry are
shown in Fig. 10.1. These parameters can be adopted as design variables:
X ¼ f D1 D2 Dn t1 t2 tn g ð10:3Þ
Design constraints are divided into some groups including the operational, stress,
and stability constraints. The operational constraint is the restricted rotation at the
top of pole structure that is limited to 1.5 . The stress constraint is considered
according to ASICE-LRFD [15] manual. The constraint on the local stability of the
cross-section is achieved as follows:
Di E Di
0:11 ) 96:25 ð10:4Þ
ti Fy ti
where E and Fy are the modulus of elasticity and minimum yield stress of the
material, respectively. Here, it is assumed that the material type is st-37 (E ¼ 210
GPa, Fy ¼ 240 MPa, and ρ ¼ 7928.5 kg/m3).
The cost function is the weight of the monopole structure, which may be expressed
as
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10.2 Monopole Structure Optimization Problem 189
X
n X
n X
n
f ðX Þ ¼ ρV i ¼ ρAi li ¼ ρð2πr i ti Þli ð10:5Þ
i¼1 i¼1 i¼1
where ρ is the weight per volume of monopole material and Vi, Ai, and li are the
volume, cross-sectional area, and length of ith part of monopole structure,
respectively.
In this study, TIA-EIA222F [6] specifications are used for considering the wind and
ice loading and their influence on structures. The applied loads on the monopole
structures consist of the vertical and horizontal loads, which are described in the
following subsections.
The most effective vertical loads, which should be considered in analysis process,
consist of the self-weight of structure, the weight of ice, and the weight of appur-
tenance (i.e., dish, light rod, and cable). For considering the load of ice weight, it is
assumed that the type of ice is solid and its density (ρice) is equal to 897.043 kg/m3
and thickness of attached ice on structure (tice) is 0.0127 m (0.5 in). Thus, the weight
of ice on unit length of ith part of pole structure (Wice
i ) is calculated as
where Si and Di are the circumference and diameter of cross section of the ith part.
The (Wice
i ) load is a uniform load which is vertically assigned to the ith part.
In the load case of attached appurtenance weight at the top of pole structure, the
weight of feedle cable of monopole is assumed as 2721.6 kg. The weight of dish and
light rod with and without ice weight are also assumed as in Table 10.1. It should be
noted that these concentrated loads are assigned to the top point of the pole
structure.
The wind load is considered as lateral load applied to the pole structure. The applied
distributed wind load to unit length of the ith part (ωwind
i ) is calculated as
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190 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
ωiwind ¼ Fi Zi ð10:7Þ
where Zi is the elevation of the center of the ith part and Fi is related to the
coefficient of wind force of the ith part which is calculated as
where Gh is the gust response factor for the fastest mile basic wind speed and it is
assumed as 1.69 for pole structures. The structure force coefficient CF is deter-
mined as 0.59 based on Table 1 of TIA/EIA-222-F. Qz is the velocity pressure and
determined as
where V is the basic wind speed of the location of the structure that is assumed as
36.1 m/s (130 km/h) and Kz is the exposure coefficient:
Also, Aei is the effective projected area of the ith part cross section in one face:
where Agi, Li, and Di are the projected area, length, and diameter of the ith part.
Moreover, the ice effect is ignored in above equation. If we consider the ice
thickness (i.e., 0.0254 m or 1 in. on the diameter of pole structure, Aei is modified as
The wind load applied to the appurtenance at the top of the pole structure is
similarly calculated. In this case, the coefficient of wind force (F) is calculated as
F ¼ Gh QzAaCa ð10:13Þ
where Aa and Ca are the projected area and force coefficients of appurtenance,
respectively. The appurtenance force coefficient (Ca) is assumed as 1.20 based on
Table 3 of TIA/EIA-222-F. The Aa is assumed as 1.45 and 1.50 m2 with and without
the effect of ice thickness on the appurtenance, respectively.
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10.3 Enhanced Colliding Bodies Optimization Algorithm 191
In this chapter, two loading combinations have been considered based on existence
of the ice load effect. Then, two loading combinations are defined:
The load combination 1 (without consideration of the ice load effect): dead load
(consisting of the self-weight of structure and weight of the appurtenance) + wind
load (consisting of the applied wind load to the face of the pole structure and
appurtenance without the ice thickness)
The load combination 2 (with consideration of the ice load effect): dead load
(consisting of the self-weight of structure, weight of the appurtenance, and ice
thickness) + wind load (consisting of the applied wind load to the face of pole
structure and appurtenance with consideration of the ice thickness)
The CBO is based on momentum and energy conservation law for one-dimensional
collision [13]. This algorithm contains a number of colliding bodies (CBs) where
each one is treated as an object with specified mass and velocity which collides with
others. After collision, each CB moves to a new position with new velocity with
respect to old velocities, masses, and coefficient of restitution. CBO starts with a set
of agents determined with random initialization of a population of individuals in the
search space. Then, CBs are sorted in an ascending order based on the values of cost
function (see Fig. 10.2a). The sorted CBs are divided equally into two groups. The
first group is the stationary group, which consists of good agents for which the
velocities before collision are zero. The second group consists of moving agents
which move toward the first group. Then, the better and worse CBs, i.e., agents with
upper fitness value, of each group collide together to improve the positions of
moving CBs and to push stationary CBs toward better positions (see Fig. 10.2b).
The change of the colliding bodies positions represent the velocities of the CBs
before collision as
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192 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
0, i ¼ 1, . . . , n
vi ¼ ð10:14Þ
xi xin , i ¼ n þ 1, . . . , 2n
where vi and xi are the velocity vector and position vector of the ith CB, respectively,
and 2n is the population size.
After the collision, the velocity of bodies in each group is evaluated using
momentum and energy conservation law and the velocities before collision. The
velocity of the CBs after the collision is
8
>
> ðmiþn þ εmiþn Þviþn
< , i ¼ 1, . . . , n
0
vi ¼ mi þ miþn ð10:15Þ
>
> ðmi εmin Þvi
: , i ¼ n þ 1, . . . , 2n
mi þ min
where vi and v0i are the velocities of the ith CB before and after the collision,
respectively, and mi is the mass of the ith CB defined as
1
fitðkÞ
mk ¼ Xn , k ¼ 1, 2, . . . , 2n ð10:16Þ
1
i¼1
fitðiÞ
where fit(i) represents the objective function value of the ith agent. Obviously, a CB
with good values exerts a larger mass and fewer moves than the bad ones. Also, for
maximizing the objective function, the term fit1ðiÞ is replaced by fit(i). ε is the
coefficient of restitution (COR) and is defined as the ratio of the separation velocity
of the two agents after collision to the approaching velocity of the two agents before
collision. In this algorithm, this index is defined to control the exploration and
exploitation rates. For this purpose, the COR decreases linearly from unit value to
zero. Here, ε is defined as
where iter is the actual iteration number and itermax is the maximum number of
iterations. Here, COR values equal to unity and zero correspond to the global and
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10.3 Enhanced Colliding Bodies Optimization Algorithm 193
local search phases, respectively. In this way a good balance between the global and
local search is achieved as the iteration number increases.
The new positions of CBs are evaluated using the generated velocities after the
collision in the position of stationary CBs:
0
xi þ rand∘vi , i ¼ 1, . . . , n
xinew ¼ 0 ð10:18Þ
xin þ rand∘vi , i ¼ n þ 1, . . . , 2n
where xnew
i and v0i are the new position and the velocity after the collision of the ith
CB, respectively.
In order to improve the CBO to obtain faster and more reliable solutions, ECBO
was developed which uses a memory to save a number of historically best CBs and
also utilizes a mechanism to escape from local optima [14]. The steps of this
technique are given as follows:
Level 1: Initialization
Step 1: The initial positions of all the CBs are determined randomly in the search
space.
Level 2: Search
Step 2: The value of mass for each CB is evaluated according to Eq. (10.16).
Step 3: Colliding memory (CM) is utilized to save a number of historically best CB
vectors and their related mass and objective function values. Solution vectors
which are saved in CM are added to the population and the same number of
current worst CBs are removed. Finally, CBs are sorted according to their
masses in a decreasing order.
Step 4: CBs are divided into two equal groups: (i) stationary group and (ii) moving
group (Fig. 10.2).
Step 5: The velocities of stationary and moving bodies before collision are evalu-
ated by Eq. (10.14).
Step 6: The velocities of stationary and moving bodies after the collision are
evaluated using Eq. (10.15).
Step 7: The new position of each CB is calculated by Eq. (10.18).
Step 8: A parameter like Pro within (0, 1) is introduced, which specifies whether a
component of each CB must be changed or not. For each colliding body, Pro is
compared with rni (i ¼ 1, 2, . . ., n) which is a random number uniformly distrib-
uted within (0, 1). If rn < Pro, one dimension of the ith CB is selected randomly
and its value is regenerated as follows:
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194 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
xij ¼ xj, min þ random: xj, max xj, min ð10:19Þ
where xij is the jth variable of the ith CB and xj,min and xj,max are the lower and upper
bounds of the jth variable, respectively. In order to protect the structures of CBs,
only one dimension is changed.
X ¼ f D1 D2 D3 D4 D5 t1 t2 t3 t4 t5 g ð10:20Þ
Design variables can be selected from a discrete list of available values set D ¼
{20, 21, 22, . . ., 89, 90} cm and t ¼ {0.4, 0.45, 0.5, 0.6, 0.8, 0.9, 1} cm, which have
78 discrete values.
Table 10.2 compares the results obtained by both algorithms with engineering
design values, for which the appropriate values are determined by the author using
trial–error method [16]. The constraint values are also shown in Table 10.2; it can
be seen that all constraints of the results of both algorithms are satisfied. Moreover,
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10.4 Design Examples 195
Table 10.2 Optimum design variables (cm) for the 30 m high monopole using different methods
Design variables Engineering design CBO ECBO
D5 40 38 38
D4 47 50 55
D3 60 57 59
D2 70 73 69
D1 80 75 76
t5 0.45 0.6 0.4
t4 0.5 0.6 0.6
t3 0.8 0.6 0.8
t2 0.8 0.8 0.8
t1 1 1 0.8
Weight (kg) 3329.4 3253.4 3123.1
Rotation 1.3454 1.3469 1.3499
Maximum stress ratio 0.4194 0.4416 0.4574
Maximum (D/t) 94.00 95.00 95.00
Rotation: rotation at top pole structure (degree)
the evolution process of best fitness values obtained by both algorithms are shown
in Fig. 10.3.
X ¼ f D1 D2 D3 D4 D5 D6 t1 t2 t3 t4 t5 t6 g ð10:21Þ
Table 10.3 compares the results obtained by both algorithms with engineering
design values. All of the constraints for the designs obtained by both algorithms are
satisfied as the first example. Moreover, the evolution process of best fitness values
obtained by both algorithms are shown in Fig. 10.4.
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196 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
(a) 5500
5000 CBO
ECBO
Weight (kg)
4500
4000
3500
3000
0 50 100 150 200
Iteration
(b)
4400
4200 CBO
ECBO
4000
Weight (kg)
3800
3600
3400
3200
3000
10 60 110 160
Iteration
Fig. 10.3 Comparison of the convergence rates between the two algorithms for the first example.
(a) All iterations, (b) 10–200 iterations [1]
In this section, the results obtained in the examples will be discussed. Firstly, it
should be noted that optimization of monopole structures is a non-convex and
nonlinear optimization problem, because the stiffness and applied loads [consisting
of the self-weight, ice, and wind load as described in Eqs. (10.6–10.13)] simulta-
neously increase with increasing the cross-sectional diameters of parts.
Tables 10.2 and 10.3 compare the results obtained using the CBO and ECBO
algorithms with the engineering design ones for both examples, respectively. As
discussed before and shown in these tables, the constraints of the final designs of
both algorithms are satisfied, and therefore these results could be compared with the
engineering design. As anticipated the results obtained using both algorithms are
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10.6 Concluding Remarks 197
Table 10.3 Optimum design variables (cm) for the 36 m high monopole using different methods
Design variables Engineering design CBO ECBO
D6 43 40 39
D5 57 56 56
D4 66 65 64
D3 73 74 74
D2 75 76 76
D1 85 86 86
t6 0.5 0.45 0.45
t5 0.6 0.60 0.60
t4 0.8 0.80 0.80
t3 0.8 0.80 0.80
t2 0.8 0.80 0.80
t1 1 1 0.90
Weight (kg) 4608.55 4557.59 4430.80
Rotation 1.4115 1.4449 1.4951
Maximum stress ratio 0.6247 0.6060 0.6041
Maximum (D/t) 95.00 95.00 95.00
Rotation: rotation at top pole structure (degree)
better than the engineering design for both examples. Moreover, the results
obtained by the ECBO algorithm are better than those of CBO using the same
number of objective function evaluations.
It can be seen from Figs. 10.3 and 10.4, though the CBO algorithm is consider-
ably faster in the early optimization iterations, the ECBO algorithm has converged
to a significantly better design in the latter optimization iterations without being
trapped in local optima.
An efficient optimization method is proposed for optimal design of the steel circular
stepped monopole structures, based on CBO and ECBO algorithms. The CBO
mimics the laws of collision between objects. The simple implementation and
parameter independency are definite strength points of CBO. In the ECBO, some
strategies have been utilized to promote the exploitation ability of the CBO. In
order to find the optimal cross-sectional sizes of monopole structure, the weight of
monopole and cross-sectional sizes are respectively defined as objective function
and variables in the optimization process. Then, the cross-sectional sizes are
selected based on optimization algorithms from available discrete variables.
The validity and efficiency of the proposed method are shown through two test
problems. The results of the proposed algorithms are compared to those of the
engineering design values. The results indicate that both algorithms could decrease
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198 10 Optimal Design of the Monopole Structures Using the CBO and ECBO Algorithms
(a) 12000
11000
CBO
10000 ECBO
9000
Weight (kg)
8000
7000
6000
5000
4000
0 50 100 150 200
Iteration
(b) 5200
5000 CBO
ECBO
Weight (kg)
4800
4600
4400
4200
10 60 110 160
Iteration
Fig. 10.4 Comparison of the convergence rates between the two algorithms for the second
example. (a) All iterations, (b) 10–200 iterations [1]
the weight of engineering design monopole structures without causing any viola-
tions. Moreover, the ECBO algorithm clearly outperforms the CBO algorithm with
the same computational time. This indicates the importance of selecting the effec-
tive optimization algorithm in this problem. Future researches can investigate
problems such as optimization of other types of monopole structures using recently
developed metaheuristic optimization algorithms.
References
Kaveh A, Mahdavi VR, Kamalinejad M (2016) Optimal design of the monopole structures using
CBO and ECBO algorithms. Periodica Polytech Civil Eng (Published online)
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5. Hu J (2011) A study on cover plate design and monopole strengthening application. Thin Wall
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6. TIA Telecommunication Industry Association (1996) TIA/EIA-222-F(TIA-Rev. F): structural
standards for steel antenna towers and antenna supporting structures. Arlington, VA
7. Kaveh A (2014) Advance in metaheuristic algorithms for optimal design of structures.
Springer, Wien, New York, NY
8. Kaveh A, Mahdavi VR (2015) Colliding bodies optimization; extensions and applications.
Springer International Publishing, Switzerland
9. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings
of the sixth international symposium on micro machine and human science, Nagoya, Japan
10. Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of
cooperating agents. IEEE Trans Syst Man Cybern B 26(1):29–41
11. Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search.
Acta Mech 213:267–289
12. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61
13. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 139:18–27
14. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
15. American Institute of Steel Construction (AISC) (2005) Steel construction manual (AISC
specification), 13th edn. AISC, Chicago, IL
16. Kamalinejad M (2005) Constructing of cell towers for the IRANCELL operator. PARTO
SANAT VIRA company, registration no. 315839, Limited liability company
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Chapter 11
Damage Detection in Skeletal Structures
Based on CSS Optimization Using Incomplete
Modal Data
11.1 Introduction
It is well known that damaged structural members may alter the behavior of the
structures considerably. Careful observation of these changes has often been
viewed as a means to identify and assess the location and severity of damages in
structures. Among the responses of a structure, natural frequencies and natural
modes are both relatively easy to obtain and independent from external excitation
and, therefore, can be used as a measure of the structural behavior before and after
an extreme event which might have led to damage in the structure. This chapter
applies charged system search algorithm to the problem of damage detection using
vibration data. The objective is to identify the location and extent of multi-damage
in a structure. Both natural frequencies and mode shapes are used to form the
required objective function. To moderate the effect of noise on measured data, a
penalty approach is applied. A variety of numerical examples including beams,
frames, and trusses are examined. The results show that the present methodology
can reliably identify damage scenarios using noisy measurements and incomplete
data [1].
During the past two decades, structural damage identification has gained increas-
ing attention from the scientific and engineering communities, since damage that is
not detected and not repaired may lead to catastrophic structural failure. Former
methods of damage identification either visual or localized experimental methods
require that the vicinity of the damage is known and accessible. Hence, the
vibration-based damage identification method as a global damage identification
technique is developed to overcome these difficulties. The basic idea of vibration-
based damage methods is that modal parameters (notably frequencies, mode
shapes, and modal damping) are functions of the physical properties of the structure
(mass, damping, and stiffness). Therefore, changes in the physical properties will
cause changes in the modal properties [2].
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11.3 Optimization Algorithm 203
The objective function is based on natural frequencies and mode shapes and is
given by Eq. (11.1). Due to measurement noise, tendency will always be to find
damage at most of the elements [17]. Thus, a penalty is introduced to weigh against
an increased number of damage sites:
0 2 1
Xr
B j ω m
ω j
a
C
Eω ¼ @ 2 A ð11:3Þ
j¼1 ωjm
where ωm j and ωj are the jth measured and analytical natural frequencies of the
a
damaged structure, respectively; ϕmj and ϕj are the measured and analytical values
a
of the jth mode shapes, respectively; r is the number of measured modes; and β is a
penalty factor which is related to the type of structure and the closeness of the
measured data and the exact data. Here, penalty is the number of damaged elements
in the analytical model.
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204 11 Damage Detection in Skeletal Structures Based on CSS Optimization Using. . .
Level 1: Initialization
Step 1: Initialization. Initialize the parameters of the CSS algorithm. Initialize an
array of charged particles (CPs) with random positions. The initial velocities of
the CPs are taken as zero. Each CP has a charge of magnitude (q) defined
considering the quality of its solution as
fitðiÞ fitworst
qi ¼ ð11:4Þ
fitbest fitworst
where fitbest and fitworst are the best and the worst fitness of all the particles
respectively, and fit(i) represents the fitness of agent i. The separation distance rij
between two charged particles is defined as
Xi Xj
r ij ¼ ð11:5Þ
ðXi þXj Þ
2 Xbest þ ε
where Xi and Xj are the positions of the ith and jth CPs, respectively, Xbest is the
position of the best current CP, and ε is a small positive value to avoid singularities.
Step 2: CP ranking. Evaluate the magnitudes of the fitness function for the CPs,
compare with each other, and sort them in increasing order.
Step 3: CM creation. Store the number of the first CPs equal to charged memory
size (CMS) and their related values of the fitness functions in the charged
memory (CM).
Level 2: Search
Step 1: Attracting force determination. Determine the probability of moving each
CP toward the others considering the following probability function:
8
< iter
1 , fitðiÞ > fitðjÞ _ 0:02 1 > rand
pmji ¼ itermax ð11:6Þ
:
0, else,
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11.3 Optimization Algorithm 205
Fj
Xj, new ¼ rand j1 ∙ ka ∙ ∙ Δt2 þ rand j, 2 ∙ kv ∙ V j, old ∙ Δt þ Xj, old ð11:8Þ
mj
where randj2 and randj2 are two random numbers uniformly distributed in the range
(1,0); mj is the mass of the CPs, which is set to unity in this chapter; Δt is the time
step, which is set to 1; ka is the acceleration coefficient; and kv is the velocity
coefficient to control the influence of the previous velocity. In this chapter kv and ka
are taken as
iter
k a ¼ ca 1 þ ð11:10Þ
itermax
iter
k v ¼ cv 1 ð11:11Þ
itermax
where ca and cv are two constants to control the exploitation and exploration of the
algorithm, iter is the iteration number, and itermax is the maximum number of
iterations.
Step 3: CP position correction. If each CP exits from the allowable search space,
correct its position.
Step 4: CP ranking. Evaluate and compare the values of the fitness function for the
new CPs, and sort them in an increasing order.
Step 5: CM updating. If some new CP vectors are better than the worst ones in the
CM, in terms of their objective function values, include the better vectors in the
CM and exclude the worst ones from the CM.
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206 11 Damage Detection in Skeletal Structures Based on CSS Optimization Using. . .
each agent can affect on the moving of the subsequent CPs, while in the standard
CSS unless an iteration is completed, the new positions cannot be utilized. Due to
using the information obtained by the CPs immediately after creation, this modifi-
cation enhances the intensification of the algorithm [19].
In this section, the efficiency and effectiveness of the proposed methods are
evaluated through some numerically simulated damage identification tests using
incomplete modal data. A continuous beam, a three-story and three-span plane
frame, and a two- and three-dimensional truss are considered with two different
damage scenarios for each of them. Due to the stochastic nature of the metaheuristic
algorithms for each scenario, the algorithm is run ten times and the solution with the
lowest cost is selected as the ultimate damage scenario. The mode shapes are
measured with less accuracy than the natural frequencies. In order to simulate the
conditions of a real test, the measured parameters are numerically perturbed by 1 %
for natural frequencies and 3 % for mode shapes to consider the presence of the
noise.
For the first example, a continuous beam depicted in Fig. 11.1 is considered. Beam
length is equally divided into 26 elements with a uniform section (IPE240). The
area of cross section and moment of inertia of the simulated beam are 39.1 cm2 and
3892 cm4, respectively. The modulus of elasticity and the material density are
200 GPa and 7780 kg/m3, respectively. The first six natural frequencies and mode
shapes of the structure are used to form the objective function. Figures 11.2 and
11.3 represent the damage states found by both optimization algorithms with the
actual damage states in different scenarios.
The frame with three spans and three stories depicted in Fig. 11.4 is considered as
the second example. The sections used for the beams and columns are IPE240 and
IPE300, respectively. The modulus of elasticity and material density are identical to
those of the previous model. The first six natural frequencies and six mode shapes of
the structure are utilized to form the objective function. Figures 11.5 and 11.6
represent the damage states found by both optimization algorithms with the actual
damage states in different scenarios.
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11.4 Numerical Examples 207
12 3 45 6 7 89 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Damage Identification
30
Real
CSS
25
Damage Percentage
ECSS
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Number of Element
Fig. 11.2 Damage detection results of the algorithms for the beam (scenario I)
Damage Identification
60
Real
CSS
50
Damage Percentage
ECSS
40
30
20
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Number of Element
Fig. 11.3 Damage detection results of the algorithms for the beam (scenario II)
1 2 3
300 cm 4 5 6 7
8 9 10
300 cm 11 12 13 14
15 16 17
300 cm 18 19 20 21
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208 11 Damage Detection in Skeletal Structures Based on CSS Optimization Using. . .
Damage Identification
30
Real
CSS
25
Damage Percentage
ECSS
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Number of Element
Fig. 11.5 Damage detection results of the algorithms for the three-span two-story frame (scenario I)
Damage Identification
30
Real
CSS
Damage Percentage
25
ECSS
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Number of Element
Fig. 11.6 Damage detection results of the algorithms for the three-span two-story frame (scenario II)
As the third example, a statically indeterminate truss bridge shown in Fig. 11.7 is
considered. The area of cross section for all elements is taken as 10 cm2. The
modulus of elasticity and material density are the same as the previous model. The
first five natural frequencies and mode shapes of the structure are used to form the
objective function. Figures 11.8 and 11.9 represent the damage states found by both
optimization algorithms with the actual damage states in different scenarios.
A space truss is considered as the last example. The geometry, element numbering,
and material properties are shown in Fig. 11.10. The first six natural frequencies
and mode shapes of the structure are utilized to form the objective function.
Figures 11.11 and 11.12 represent the damage states found by both optimization
algorithms with the actual damage states in different scenarios.
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11.5 Concluding Remarks 209
40 cm 3 4
60 cm
2 5
21
23
19
15
25
160 cm
14 16 6
1 13 20 22
18 24 17
7 8 9 10 11 12
6@200 cm
Damage Identification
35
Real
30 CSS
Damage Percentage
ECSS
25
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number of Element
Fig. 11.8 Damage detection results of the algorithms for the planar truss (scenario I)
Damage Identification
35
Real
30 CSS
Damage Percentage
ECSS
25
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number of Element
Fig. 11.9 Damage detection results of the algorithms for the planar truss (scenario II)
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210 11 Damage Detection in Skeletal Structures Based on CSS Optimization Using. . .
Damage Identification
35
Real
30 CSS
Damage Percentage
ECSS
25
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number of Element
Fig. 11.11 Damage detection results of the algorithms for the space truss (scenario I)
Damage Identification
35
Real
30 CSS
Damage Percentage
ECSS
25
20
15
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number of Element
Fig. 11.12 Damage detection results of the algorithms for the space truss (scenario II)
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References 211
References
1. Kaveh A, Maniat M (2014) Damage detection in skeletal structures based on charged system
search optimization using incomplete modal data. Int J Civil Eng IUST 12(2):291–298
2. Doebling Scott W, Farrar CR, Prime MB (1998) A summary review of vibration-based damage
identification methods. Shock Vib Dig 30:91–105
3. Perera R, Torres R (2006) Structural damage detection via modal data with genetic algorithms.
J Struct Eng 132:1491–1501
4. Laier JE, Morales JD (2009) Improved genetic algorithm for structural damage detection.
Comput Struct Eng 91:833–839
5. Miguel L, Miguel L, Kaminski J Jr, Riera J (2012) Damage detection under ambient vibration
by harmony search algorithm. Expert Syst Appl 39(10):9704–9714
6. Kang F, Li JJ, Xu Q (2012) Damage detection based on improved particle swarm optimization
using vibration data. Appl Soft Comput 12(8):2329–2335
7. Majumdar A, Kumar Maiti D, Maity D (2012) Damage assessment of truss structures from
changes in natural frequencies using ant colony optimization. Appl Math Comput
218:9759–9772
8. Fan W, Qiao P (2011) Vibration-based damage identification methods: a review and compar-
ative study. Struct Health Monit 10:83–111
9. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings
of the sixth international symposium on micro machine and human science, Nagoya, Japan
10. Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of
cooperating agents. IEEE Trans Syst Man Cybern B26:29–41
11. Geem ZW, Kim JH, Loganathan GV (2001) A new heuristic optimization algorithm: harmony
Search. Simulation 76:60–68
12. Erol OK, Eksin I (2006) New optimization method: Big Bang–Big Crunch. Adv Eng Softw
37:106–111
13. Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search.
Acta Mech 213:267–289
14. Kaveh A, Motie Share MA, Moslehi M (2013) A new metaheuristic algorithm for optimiza-
tion: magnetic charged system search. Acta Mech 224:85–107
15. Kaveh A, Khayatazad M (2012) A new metaheuristic method: ray optimization. Comput Struct
112–113:283–294
16. Kaveh A, Farhoudi N (2013) A new optimization method: dolphin echolocation. Adv Eng
Softw 59:53–70
17. Friswell MI, Penny JET, Garvey SD (1998) A combined genetic and Eigen-sensitivity
algorithm for the location of damage in structures. Comput Struct 69:547–556
18. Kaveh A, Talatahari S (2011) Optimization of large-scale truss structures using modified
charged system search. Int J Optim Civil Eng 1:15–28
19. Kaveh A, Talatahari S (2011) An enhanced charged system search for configuration optimi-
zation using the concept of fields of forces. Struct Multidiscip Optim 43:339–351
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Chapter 12
Modification of Ground Motions Using
Enhanced Colliding Bodies Optimization
Algorithm
12.1 Introduction
In this chapter a simple and robust approach is presented for spectral matching of
ground motions utilizing the wavelet transform and an improved metaheuristic
optimization technique. For this purpose, wavelet transform is used to decompose
the original ground motions to several levels, where each level covers a special
range of frequency, and then each level is multiplied by a variable. Subsequently,
the enhanced colliding bodies optimization (ECBO) technique is employed to
calculate the variables such that the error between the response and target spectra
is minimized. The application of the proposed method is illustrated through mod-
ifying 12 sets of ground motions [1].
Recent aseismic code regulations recommend the use of linear or nonlinear
dynamic time history analyses for design of irregular, high rise, and important
structures due to the increased capabilities of the commercial software to account
for the potential inelastic behavior of structural systems under seismic time histo-
ries. These acceleration time histories can be achieved either by using a set of real
recorded earthquake accelerograms associated with historical seismic events, or
utilizing an ensemble of numerically simulated earthquake signals. In the latter
approach, one can make pure artificial records and filter them according to the site
characteristics or to reconstruct the real record so that its spectrum fits the target
standard [2]. Obviously finding suitable methods for reconstructing or modifying
realistic ground motions is an important and challenging problem.
The main objective of the reconstruction/modification of ground motions is to
modify a given set of ground motions such that these response spectrums become
compatible with a specified design spectrum. For this purpose, various time or
frequency-domain methods are used. The time-domain methods manipulate only
the amplitude of the recorded ground motions, while the frequency-domain
approaches operate the frequency contents and phasing of actual ground motions
in order to match with the design spectrum. During the last two decades, a number
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12.2 Spectral Matching Problem According to Eurocode-8 215
The elastic acceleration response spectrum, Sa(T ), for oscillators with 5 % ratio of
critical damping and natural period, T, is defined by the European seismic code
provisions (CEN [11]) as
8
>
> αg S 1 þ
1:5T
0 T TB
>
>
>
> TB
>
> TB T Tc
< 2:5αg S
Sa ð T Þ ¼ TC ð12:1Þ
>
> 2:5αg S TC T TD
>
> T
>
>
>
> T T
: 2:5αg S C 2 D T D T 4s
T
where S is the soil factor, TB and TC are the limiting periods of the constant spectral
acceleration branch, TD defines the beginning of the constant displacement response
range of the spectrum, and ag is the design ground acceleration on type A ground,
which is defined according to the seismic hazard. In this study, ag is chosen as
0.35 g.
The values of the periods TB, TC, and TD and the soil factor S describing the
shape of the elastic response spectrum depend on the ground type. In Table 12.1, the
specific values that determine the spectral shapes for Type 1 spectra are listed, and
the resulting spectra is normalized by ag and plotted in Fig. 12.1.
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216 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
Fig. 12.1 Elastic response spectra for different site soil classes, based on the EC8
smaller than the value of agS for the site in question; and (3) in the range of periods
between 0.2Tn and 2Tn, where Tn is the fundamental period of the structure in the
direction where the accelerogram is applied, no value of the mean 5 % damping
elastic spectrum calculated from all time histories should be < 90 % of the
corresponding value of the 5 % damping elastic response spectrum.
Moreover, the code allows the consideration of the mean effect on the structure,
rather than the maximum effect if at least seven nonlinear time history analyses are
performed.
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12.3 Wavelet Transform 217
X
1
Dj ðtÞ ¼ cDj ðkÞψ j, k dk ð12:2Þ
k¼1
where ψ j is the wavelet function, k is the translation parameter, and cDj(k) is the
wavelet coefficient at level j which is defined as
ð
1
X
1
A j ðt Þ ¼ cAj ðkÞφj, k dk ð12:4Þ
k¼1
where φj is the scaling function and cAj(k) is the scaling coefficient at level j which
is defined as
ð
1
In this chapter for decomposing the signals, Daubechies wavelet and scaling
function of order 10 (db-10) are used [19]. Finally, the signal f(t) can be represented by
X
f ðtÞ ¼ An ðtÞ þ Dj ðtÞ ð12:6Þ
jn
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218 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
In wavelet transformation, scaling and wavelet functions are used. These are
related to low-pass and high-pass filters, respectively. A wavelet function can also
be represented as
1 t 2j k
ψ j, k ðtÞ ¼ pffiffiffiffi ψ ð12:7Þ
2j 2j
where Δt is the time step of the signal f(t) (Refs. [20, 21]).
An iterative method is used for solving spectral matching problem that is based on
the work of Mukherjee and Gupta [7]. In this method, first an ordinary ground
motion is decomposed using wavelet transform, and detailed signals are deter-
mined. Then, the ground motion is modified by scaling each of the detailed signals
(Dj) up/down based on the amplification/reduction required to reach target spectral
ordinates in the period band corresponding
to that time history. Thus, in the ith
i
iteration, the detailed signals Dj are modified for level j to the modified detailed
i þ1
ð T2
½Sa ðT ÞTarget dT
Diþ1
j ¼ Dji ð T2 T1
ð12:11Þ
½PSAðT Þcalculated dT
T1
where T1 and T2 are the period bounds on the range of level j [Eq. (12.10)]. Finally,
a modified ground motion is constructed using Eq. (12.6). The disadvantages of this
method can be mentioned as (i) it modifies only one ground motion, (ii) it cannot
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12.4 The Proposed Methodology 219
X
n
f m ðt Þ ¼ αj Dj þ αnþ1 An ð12:12Þ
j¼1
where Dj and An are the detailed and approximate signals at levels j and n,
respectively, and αj is the jth modified value. In fact, this value is a variable in
the optimization process. The number of optimization variables is equal to n + 1
multiplied by the number of ground motions, and in the present chapter, this is
equal to 9 * 3 ¼ 27.
Step 4. Creation of the response spectrum: In this step, the response pseudo-
acceleration spectrums of the modified ground motions is determined. As men-
tioned before based on Eurocode-8, when a set of 3 through 6 ground motions is
used, the structural engineer should use the maximum response value instead of
the mean response value. Hence, the response spectrum of ground motions
should be calculated as
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220 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
2π
PSAðω; ξÞ ¼ ω2 max ðjxðtÞjÞ, ξ ¼ 5%, ω¼ ð12:14Þ
t T
where ω, ζ, and fm(t) are the fundamental frequency, the damping coefficient of
the single degree of freedom system, and the earthquake ground acceleration,
respectively.
Step 5. Determination of the penalty function: In this chapter penalty method is
utilized to satisfy the code requirements:
Penalty ¼ P1 þ P2 þ P3 ð12:16Þ
P1 ¼ max 0, max ð0:9*Sa ðT i Þ PSAðT i ÞÞ , 0:2T n T i 2T n ð12:17Þ
i
P3 ¼ max 0, max ðαi Þ , i ¼ 1, 2, . . . , 27 ð12:19Þ
i
where X is the vector of the optimization variables [i.e., the modified values in
Eq. (12.12)], λ is a large number which is selected to magnify the penalty effects,
and Err is calculated using Eq. (12.21) as the response spectrum becomes close
to the target spectrum:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u N
u1 X
ErrðXÞ ¼ 100*t ðlogðSa ðT i ÞÞ logðPSAðT i ÞÞÞ2 ð12:21Þ
N i¼1
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12.5 Enhanced Colliding Bodies Optimization Algorithm 221
where N is the number of specified periods. Here, 500 period points are consid-
ered in the range [0–5] s with period steps of 0.01 s.
Step 7. Termination criterion: The optimization process is repeated starting with
Step 3 until the maximum number of iterations as a termination criterion is
attained.
Step 8. Correction of baseline: The velocity and displacement time history of
reconstructed ground accelerations do not become unrealistic due to systematic
low-frequency errors. Hence, the baseline correction of the modified
accelerograms is needed for this purpose.
The flowchart of this method is shown in Fig. 12.3.
The CBO is based on momentum and energy conservation law for one-dimensional
collision (Kaveh and Mahdavi [23]). This algorithm contains a number of colliding
bodies (CB) where each one is treated as an object with specified mass and velocity
which collides with others. After collision, each CB moves to a new position with
new velocity with respect to previous velocities, masses, and coefficient of restitu-
tion. CBO starts with a set of agents determined with random initialization of a
population of individuals in the search space. Then, CBs are sorted in an ascending
order based on the values of their cost functions (see Fig. 12.4a). The sorted CBs are
divided equally into two groups. The first group is stationary and consists of good
agents. This set of CBs is stationary and their velocity before collision is zero. The
second group consists of moving agents which move toward the first group. Then,
the better and worse CBs, i.e., agents with upper fitness values of each group,
collide together to improve the positions of the moving CBs and to push stationary
CBs toward better positions (see Fig. 12.4b). The change of the body position
represents the velocity of the CBs before collision as
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222 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
0, i ¼ 1, . . . , n
vi ¼ ð12:22Þ
xi xin , i ¼ n þ 1, . . . , 2n
where, νi and xi are the velocity vector and position vector of the ith CB, respec-
tively. 2n is the number of population size.
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12.5 Enhanced Colliding Bodies Optimization Algorithm 223
After the collision, the velocity of bodies in each group is evaluated using
momentum and energy conservation law and the velocities before collision
[Eq. (12.22)]. The velocity of the CBs after the collision becomes
8
>
> ðmiþn þ εmiþn Þviþn
< , i ¼ 1, . . . , n
0
vi ¼ mi þ miþn ð12:23Þ
>
> ðmi εmin Þvi
: , i ¼ n þ 1, . . . , 2n
mi þ min
where vi and vi0 are the velocities of the ith CB before and after the collision,
respectively; mi is the mass of the ith CB defined as
1
fitðkÞ
mk ¼ Xn , k ¼ 1, 2, . . . , 2n ð12:24Þ
1
i¼1
fitðiÞ
where fit(i) represents the objective function value of the ith agent. Obviously a
CBs with better objective function values will be assigned with larger mass values.
Also, for maximizing the objective function, the term 1/fit(i) is replaced by fit(i). ε is
the coefficient of restitution (COR) and is defined as the ratio of the separation
velocity of the two agents after collision to approach velocity of two agents before
collision. In this algorithm, this index is defined to control the exploration and
exploitation rates. For this purpose, the COR decreases linearly from unit value to
zero. Here, ε is defined as
iter
ε¼1 ð12:25Þ
iter max
where iter is the actual iteration number and itermax is the maximum number of
iterations. Here, COR is equal to unity and zero representing the global and local
search, respectively. In this way a good balance between the global and local search
is achieved by increasing the iteration number.
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224 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
The new positions of CBs are evaluated using the generated velocities after the
collision:
0
xi þ rand∘vi , i ¼ 1, . . . , n
xinew ¼ 0 ð12:26Þ
xin þ rand∘vi , i ¼ n þ 1, . . . , 2n
where xinew and xi are the new position and the velocity after the collision of the
ith CB, respectively.
In order to improve the CBO to obtain faster and more reliable solutions, ECBO is
developed which uses memory to save a number of historically best CBs and also
utilizes a mechanism to escape from local optima (Kaveh and Ilchi Ghazaan [18]).
The steps of this technique are as follows:
Level 1: Initialization
Step 1: The initial positions of all the CBs are determined randomly in the search
space.
Level 2: Search
Step 1: The value of mass for each CB is evaluated according to Eq. (12.24).
Step 2: Colliding memory (CM) is utilized to save a number of historically best CB
vectors and their related mass and objective function values. Solution vectors
which are saved in CM are added to the population, and the same number of
current worst CBs are removed. Finally, CBs are sorted according to their
masses in a decreasing order.
Step 3: CBs are divided into two equal groups: (i) stationary group and (ii) moving
group (Fig. 12.4).
Step 4: The velocities of stationary and moving bodies before collision are evalu-
ated by Eq. (12.22).
Step 5: The velocities of stationary and moving bodies after the collision are
evaluated using Eq. (12.23).
Step 6: The new position of each CB is calculated by Eq. (12.26).
Step 7: A parameter like Pro within (0, 1) is introduced, which specifies whether
a component of each CB must be changed or not. For each colliding body, Pro
is compared with rni (i ¼ 1, 2, . . ., n) which is a random number uniformly
distributed within (0, 1). If rn < Pro, one dimension of the ith CB is selected
randomly, and its value is regenerated as follows:
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12.6 Numerical Examples 225
xij ¼ xj, min þ random: xj, max xj, min ð12:27Þ
where xij is the jth variable of the ith CB and xj,min and xj,max are the lower and
upper bounds of the jth variable, respectively. In order to protect the structures of
CBs, only one dimension is changed.
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226 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
In the optimization process of all the cases, the CBO and ECBO algorithms
are used to provide a comparison between these two algorithms. In these cases,
the number of agents is set as 30. The maximum number of iterations is also
considered as 300. As mentioned before, the well-known penalty approach
is used for satisfying the code requirements. Comparisons are made through the
error between the target spectrum and modified maximum response spectrums
[Eq. (12.21)]. The algorithms are also coded in MATLAB.
Figures 12.5 and 12.6 display the original and modified acceleration and
the displacement time histories of the SetA-1, respectively. From these figures it
can be seen that the frequency contents of the modified acceleration time histories
are different with those of original ones. In this case, comparing the actual and
modified accelerograms, it can be seen that the modified acceleration and dis-
placement time histories of the Anza and Kocaeli earthquakes are modified more
than the Loma Prieta earthquake. The modified displacement time histories are
also realistic due to the use of the baseline correction in the last step of proposed
method.
The maximum response spectrums of the SetA-1 original and modified ground
motions obtained by both algorithms for three fundamental periods and target
spectrum are shown in Fig. 12.7. The 90 % design spectrum (the red dashed lines)
and the period ranges of interest (the vertical blue dashed lines) are also displayed
as these are the spectral amplitude limits specified by the Eurocode-8. It can be seen
the maximum response spectrum of the original accelerograms is far away from the
target spectrum, and it falls below the 90 % design spectrum within the period limits
as well. While the maximum response spectrums of the modified accelerograms
have approached to the target spectrum with modification of these original ground
motions using the presented method. Also, the maximum response spectrum does
not fall below the 90 % target spectrum within the code-specific period range and
zero periods.
Figures 12.7, 12.8, 12.9, and 12.10 show the maximum response spectrums of
the modified ground motions obtained by the proposed method for the SetA-2,
SetB-1, and SetB-2 as well as three fundamental periods, respectively. Similar
results and comparisons can be obtained from these figures. Table 12.3 shows the
optimized error obtained by CBO and ECBO for all cases. As shown in this table
and Figs. 12.7, 12.8, 12.9, and 12.10, the resulted lower error leads to the
response spectrum that is close to the target spectrum. This indicates that more
suitable modification of the recorded accelerograms can be achieved using more
efficient optimization algorithms. It can be seen that the errors obtained by ECBO
are better than those obtained for the CBO algorithm, which indicates the
importance of the enhancement of the algorithm in this problem. The errors are
also decreased with increase of the fundamental period (Tn); therefore, the
recorded accelerograms can easily be modified in high fundamental periods
using the proposed method.
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12.6 Numerical Examples 227
Fig. 12.5 Original and modified acceleration time histories of (a) Anza, (b) Kocaeli, and (c)
Loma Prieta
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228 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
Fig. 12.6 Original and modified displacement time histories of (a) Anza, (b) Kocaeli, and (c)
Loma Prieta
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12.6 Numerical Examples 229
Fig. 12.7 Comparison of various maximum response spectrums of SetA-1 matched with the
target spectrum of soil class A for fundamental periods: (a) Tn ¼ 0.45, (b) Tn ¼ 0.9, (c) Tn ¼ 1.8
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230 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
Fig. 12.8 Comparison of various maximum response spectrums of SetA-2 matched with the
target spectrum of soil class A for fundamental periods: (a) Tn ¼ 0.45, (b) Tn ¼ 0.9, (c) Tn ¼ 1.8
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12.6 Numerical Examples 231
Fig. 12.9 Comparison of various maximum response spectrums of SetB-1 matched with the target
spectrum of soil class B for fundamental periods: (a) Tn ¼ 0.45, (b) Tn ¼ 0.9, (c) Tn ¼ 1.8
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232 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
Fig. 12.10 Comparison of various maximum response spectrums of SetB-2 matched with the
target spectrum of soil class B for fundamental periods: (a) Tn ¼ 0.45, (b) Tn ¼ 0.9, (c) Tn ¼ 1.8
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12.7 Concluding Remarks 233
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234 12 Modification of Ground Motions Using Enhanced Colliding Bodies. . .
References
1. Kaveh A, Mahdavi VR (2016) A new method for modification of ground motions using
wavelet transform and enhanced colliding bodies optimization. Appl Soft Comput 47:357–369
2. Bommer JJ, Acevedo AB (2004) The use of real earthquake accelerograms as input to dynamic
analysis. J Earthq Eng 8:43–91
3. Gupta ID, Joshi RG (1993) On synthesizing response spectrum compatible accelerograms. Eur
J Earthq Eng 7:25–33
4. Shrikhande M, Gupta VK (1996) On generating ensemble of design spectrum-compatible
accelerograms. J Earthq Eng 10:49–56
5. Conte JP, Peng BF (1997) Fully nonstationary analytical earthquake ground motion model. J
Eng Mech ASCE 123:15–24
6. Hancock J, Watson-Lamprey J, Abrahamson NA, Bommer JJ, Markatis A, McCoy E, Mendis
E (2006) An improved method of matching response spectra of recorded earthquake ground
motion using wavelets. J Earthq Eng 10:67–89
7. Mukherjee S, Gupta VK (2002) Wavelet-based generation of spectrum-compatible time-
histories. Soil Dyn Earthq Eng 22:799–804
8. Cecini D, Palmeri A (2015) Spectrum-compatible accelerograms with harmonic wavelets.
Comput Struct 147:26–35
9. Gao Y, Wu Y, Li D, Zhang N, Zhang F (2014) An improved method for the generating of
spectrum-compatible time series using wavelets. Earthq Spectra 30:1467–1485
10. Ghodrati Amiri G, Abdolahi Rad A, Khanmohamadi Hazaveh N (2014) Wavelet-based
method for generating nonstationary artificial pulse-like near-fault ground motions. Comput
Aided Civil Infrastruct Eng 29:758–770
11. CEN. Eurocode-8 (2003) Design provisions for earthquake resistance of structures. Part 1:
general rules, seismic actions and rules for buildings. Brussels
12. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings
of the sixth international symposium on micro machine and human science, Nagoya, Japan
13. Dorigo M, Maniezzo V, Colorni A (1996) The ant system: optimization by a colony of
cooperating agents. IEEE Trans Syst Man Cybern B 26:29–41
14. Eroland OK, Eksin I (2006) New optimization method: Big Bang–Big Crunch. Adv Eng Softw
37:106–111
15. Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search.
Acta Mech 213:267–289
16. Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2013) Mine blast algorithm: a new
population based algorithm for solving constrained engineering optimization problems. Appl
Soft Comput 13(2013):2592–2612
17. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 139:18–27
18. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
19. Daubechies I (1992) Ten lectures on wavelets. In: CBMS-NSF Conference series in applied
mathematics, Montpelier, VT
20. Ghodrati Amiri G, Bagheri A, Seyed Razaghi SA (2009) Generation of multiple earthquake
accelerograms compatible with spectrum via the wavelet packet transform and stochastic
neural networks. J Earthq Eng 13:899–915
21. Qina X, Fangb B, Tianb S, Tonga X, Wangc Z, Sua L (2014) The existence and uniqueness of
solution to wavelet collocation. Appl Math Comput 231:63–72
22. PEER NGA (2014) Strong motion database. http://peer.berkeley.edu/nga/
23. Kaveh A, Mahdavi VR (2015) A hybrid CBO–PSO algorithm for optimal design of truss
structures with dynamic constraints. Appl Soft Comput 34:260–273
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ASCE 28:04014019
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Chapter 13
Bandwidth, Profile, and Wavefront
Optimization Using CBO, ECBO, and TWO
Algorithms
13.1 Introduction
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13.2 Problem Definition 237
Let G(N,M) be a graph with members set M(|M| ¼ m) and nodes set N(|N| ¼ n) with a
relation of incidence. The degree of a node is the number of members incident with
the node, and the 1-weighted degree of a node is defined as the sum of the degrees of
its adjacent nodes. A spanning tree is a tree containing all the nodes of S. A shortest
route tree (SRTn0) rooted from a specified node (starting node) n0 is a spanning tree
for which the distance between every node nj of S and n0 is minimum, where the
distance between two nodes is defined as the number of members in the shortest
path between these nodes. A contour Cnk 0 contains all the nodes with equidistance
k from node n0. The number of contours of an SRTn0 is known as its depth, denoted
by d(SRTn0), and the highest number of nodes in a contour specifies the width of the
SRTn0. A labeling As of G assigns the set of integers {1, 2, 3, . . ., n} to the nodes of
graph G. As(i) is the label or the integer assigned to node i and each node has a
different label. The bandwidth of node i for this assignment, bw(i), is the maximum
difference of As(i) and As( j), where As( j) is the label of nodes adjacent to node i or
the number assigned to its adjacent nodes (Kaveh [3]). That is
where N(i) is the set of adjacent nodes of node i. The bandwidth of the graph G with
respect to the assignment As(i) is then
The minimum value of BW over all possible assignments is the bandwidth of the
graph:
The profile of the N N matrix related to graph G, for the assignment As(i), is
defined as
X
N
PAs ¼ bi ð13:4Þ
i¼1
where the row bandwidth, bi, for row i is defined as the number of inclusive entries
from the first nonzero element in the row to the (i + 1)th entry for this assignment.
The efficiency of any given ordering for the profile solution scheme is related to the
number of active equations during each step of the factorization process. Formally,
row j is defined to be active during elimination of column i if j i, and there exists
aik ¼ 0 with k i. Hence, at the ith stage of the factorization, the number of active
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238 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
equations is the number of rows of profile that intersect column i, which is ignored
if those rows already eliminated. Let fi denote the number of equations that are
active during the elimination of the variable xi. It follows from the symmetric
structures of the matrix that
X
N X
N
PAs ¼ fi ¼ bi ð13:5Þ
i¼1 i¼1
The nodal numbering in a priority queue is carried out through the assignment of
status, based on the numbering approach of King [9]. King’s method was general-
ized by Sloan [17], by introducing a priority queue which controls the order to be
followed in the numbering of the nodes. This algorithm comprises of two phases:
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13.2 Problem Definition 239
Pi ¼ W 1 δi W 2 Di ð13:7Þ
Active
Inactive
Post-active
Pre-active
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240 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
D i ¼ d i ci þ k i ð13:8Þ
where di is the degree of node i, ci is the number of active and post-active nodes
adjacent to node i, and ki is zero if the node i is active or post-active and unity
otherwise.
Step 4: Select the node with the highest priority among the candidate nodes and
label it.
Step 5: Repeat Steps 1–4 until all the nodes are labeled.
In Eq. (13.7) if W1 ¼ 0 and W2 ¼ 1, the node-labeling algorithm will become
similar to the one proposed by King.
As can be seen from Eq. (13.7), Sloan’s algorithm employs a linear priority function
of two graph parameters and the weights determine the importance of each param-
eter. In Sloan’s algorithm the pair W1 ¼ 1 and W2 ¼ 2 has been recommended for
the weights. However, some research results (Kaveh and Roosta [18], Rahimi
Bondarabadi and Kaveh [19]) show that for some problems, there are advantages
in using other values.
In general, the priority can be determined by a general linear function of vectors
of graph parameters and their coefficients as
X
L
Pi ¼ W i Ci ð13:9Þ
i¼1
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13.3 Metaheuristic Algorithms 241
This section includes the colliding bodies optimization algorithm, its enhanced
version, and tug of war optimization algorithm. First, a brief description of standard
CBO is provided. The ECBO is presented (Kaveh and Ilchi Ghazaan [12]), and then
a new algorithm called TWO is stated.
where fit(i) represents the objective function value of the ith CB and n is the number
of colliding bodies. In order to select pairs of objects for collision, CBs are sorted
according to their mass in a decreasing order and they are divided into two equal
groups: (i) stationary group and (ii) moving group. Moving objects collide to
stationary objects to improve their positions and push stationary objects toward
better positions (see Fig. 13.2).
The velocity of the stationary bodies before collision is zero, so
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242 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
n
vi ¼ 0, i ¼ 1 , 2 , ... , ð13:11Þ
2
0
The velocity of each moving CB after the collision (vi ) is defined by
mi εmin2 vi n n
v0i ¼ , i ¼ þ 1 , þ 2 , ... , n ð13:14Þ
mi þ min2 2 2
Here ε is the coefficient of restitution (COR) that decreases linearly from unit
to zero.
Thus, it is stated as
iter
ε¼1 ð13:15Þ
iter max
where iter is the current iteration number and itermax is the total number of
iterations for optimization process.
New positions of CBs are updated according to their velocities after the collision
and the positions of stationary CBs. Therefore, the new position of each stationary
CB is
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13.3 Metaheuristic Algorithms 243
n
xinew ¼ xi þ rand∘v0i , i ¼ 1, . . . , ð13:16Þ
2
0
where xnew
i , xi, and vi are the new position, previous position, and the velocity after
the collision of the ith CB, respectively, rand is a random vector uniformly
distributed in the range of [1, 1], and the sign “∘” denotes an element-by-element
multiplication. The new position of each moving CB is calculated by
n n
xinew ¼ xin2 þ rand∘v0i , i¼ þ 1, þ 2, . . . , n ð13:17Þ
2 2
where x0i is the initial solution vector of the ith CB. Here, xmin and xmax are the
bounds of design variables, random is a random vector in which each component
is in the interval [0, 1], and n is the number of CBs.
Step 2: Defining mass
The value of mass for each CB is evaluated according to Eq. (13.10).
Step 3: Saving
Considering a memory which saves some historically best CB vectors and their
related mass and objective function values can make the algorithm performance
better without increasing the computational cost (Kaveh and Ilchi Ghazaan
[12]). Here, a colliding memory (CM) is utilized to save a number of the best-
so-far solutions. Therefore, in this step, the solution vectors saved in CM are
added to the population, and the same numbers of current worst CBs are deleted.
Finally, CBs are sorted according to their masses in a decreasing order.
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244 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
where xij is the jth variable of the ith CB and xj,min and xj,max, respectively, are
the lower and upper bounds of the jth variable. In order to protect the structures
of CBs, only one dimension is changed. This mechanism provides opportunities
for the CBs to move all over the search space, thus providing better diversity.
Step 9: Terminating condition check
The optimization process is terminated after a fixed number of iterations. If this
criterion is not satisfied, go to Step 2 for a new round of iteration (Kaveh and Ilchi
Ghazaan [12]).
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13.3 Metaheuristic Algorithms 245
where x0ij is the initial value of the jth variable of the ith candidate solution; xj,max
and xj,min are the maximum and minimum permissible values for the jth variable,
respectively; rand is a random number from a uniform distribution in the interval
[0, 1]; and n is the number of variables.
Step 2: Evaluation and weight assignment
The objective function values for the candidate solutions are evaluated and
sorted. The best solution so far and its objective function value are saved.
Each solution is considered as a team with the following weight:
fitðiÞ fitworst
W i ¼ 0:9 þ 0:1 i ¼ 1, 2, . . . , N ð13:21Þ
fitbest fitworst
where fit(i) is the fitness value for the ith particle. The fitness value can be
considered as the penalized objective function value for constrained problems;
fitbest and fitworst are the fitness values for the best and worst candidate solutions
of the current iteration. According to Eq. (13.21), the weights of the teams range
between 0.1 and 1.
Step 3: Competition and displacement
In TWO each team competes against all the others one at a time to move to its
new position in every iteration. The pulling force exerted by a team is assumed
to be equal to its static friction force (Wμs). Hence, the pulling force between
teams i and j (Fp,ij) can be determined as max{Wiμs, Wjμs}. Such a definition
keeps the position of the heavier team unaltered.
The resultant force affecting team i due to its interaction with heavier team j in
the kth iteration can then be calculated as follows:
where Fkp;ij is the pulling force between teams i and j in the kth iteration and μk is
the coefficient of kinematic friction.
!
Frk, ij
aijk ¼ gijk ð13:23Þ
W ik μk
in which akij is the acceleration of team i toward team j in the kth iteration and gkij
is the gravitational acceleration constant which is defined as
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246 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
where Xkj and Xki are the position vectors for candidate solutions j and i in the kth
iteration. Finally, the displacement of team i after competing with team j can be
derived as
1
ΔXijk ¼ aijk Δt2 þ αk ðXmax Xmin Þ∘ð0:5 þ rand ð1; nÞÞ ð13:25Þ
2
The second term of Eq. (13.25) induces randomness into the algorithm. This
term can be interpreted as the random portion of the search space travel by team
i before it stops after the applied force is removed. Here, α is a constant chosen
from the interval [0,1]; Xmax and Xmin are the vectors containing the upper and
lower bounds of the permissible ranges of the design variables, respectively; ∘
denotes element-by-element multiplication; and rand(1, n) is a vector of uni-
formly distributed random numbers.
It should be noted that when team j is lighter than team i, the corresponding
displacement of team i will be equal to zero (i.e., ΔXkij ). Finally, the total
displacement of team i in iteration k is equal to
X
N
ΔXik ¼ ΔXijk ð13:26Þ
j¼1
The new position of team i at the end of kth iteration is then calculated as
Xkþ1
i ¼ Xik þ ΔXik ð13:27Þ
randn
xijk ¼ GBj þ GBj xijk1 ð13:28Þ
k
where GBj is the jth variable of the global best solution (i.e., the best solution so
far) and randn is the random number drawn from a standard normal distribution.
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13.4 Numerical Examples 247
There is a very slight possibility for the newly generated variable being still
outside the search space. In such cases a flyback strategy is used.
Step 5: Termination
Steps 2 through 5 are repeated until a termination criterion is satisfied (Kaveh
and Zolghadr [13]).
In this section, four finite element meshes (FEMs) are considered. Element clique
graph is a type of graph model that is employed for transferring topological
properties of finite element models into connectivity properties of graphs (Kaveh
[4]). This graph model has the same nodes as those of corresponding finite element
model, and the nodes of each element are cliqued, avoiding the multiple members
for the whole graph. The first example is a Z-shaped finite element model for shear
wall. An element clique graph of a rectangular FEM with four openings is consid-
ered in the second example. The third example is the grid model of a fan with
one-dimensional beam elements, and an H-shaped finite element grid is presented
in the fourth example. The well-known standard PSO algorithm; two new algo-
rithms, namely, the colliding bodies optimization and enhanced colliding bodies
optimization; and a recently developed method called tug of war optimization are
applied for all of three bandwidth, profile, and wavefront minimizing problems. The
results in bandwidth reduction problem are then compared to those of the four-step
algorithm of Kaveh [2] and those of Kaveh and Sharafi [14,15] in Table 13.1. The
results obtained in profile and wavefront minimizing problems with L ¼ 2 and
5 methods are compared to those of the Sloan and King’s algorithms in Tables 13.2
and 13.3, respectively.
The FEM of a shear wall with 550 nodes is considered. The element clique graph of
this model is shown in Fig. 13.3. The performance of the abovementioned
Table 13.1 Comparison of the results of different algorithms for bandwidth reduction
Kaveh and Sharafi [14,15]
4-step algorithm PSO CBO ECBO TWO 4-step ACO CSS
Example 1 28 28 28 28 28 29 29 –
Example 2 29 29 29 29 29 – – –
Example 3 18 18 18 18 18 23 23 21
Example 4 57 57 57 57 57 66 60 58
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248 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
Table 13.2 Comparison of the results of different algorithms for profile reduction
Example Algorithm W1 W2 W3 W4 W5 Profile
Example 1 Sloan 1 2 10,530
King 0 1 10,974
PSO L¼2 0.0007 0.4852 10,501
L¼5 0.0863 0.4638 0.3677 0.0034 0.9191 9280
CBO L ¼ 2 0.0043 0.4001 10,501
L¼5 0.2191 0.9551 0.6962 0.0390 0.3207 9242
ECBO L ¼ 2 0.0001 0.9881 10,501
L¼5 0.0255 0.8883 0.6256 0.0110 0.9183 9237
TWO L ¼ 2 0.0129 0.7645 10,501
L¼5 0.0404 0.8801 0.5940 0.0063 0.7426 9237
Example 2 Sloan 1 2 18,719
King 0 1 18,839
PSO L¼2 0.6645 0.9066 18,690
L¼5 0.1056 0.8858 0.4835 0.0152 0.0524 17,136
CBO L ¼ 2 0.1899 0.2566 18,689
L¼5 0.3332 0.7097 0.9412 0.0507 0.9747 17,122
ECBO L ¼ 2 0.6665 0.9228 18,581
L¼5 0.0458 0.7835 0.6332 0.0060 0.2254 17,039
TWO L ¼ 2 0.7136 0.9633 18,581
L¼5 0.0178 0.4092 0.9291 0.0024 0.5575 17,039
Example 3 Sloan 1 2 28,703
King 0 1 28,853
PSO L¼2 0.2588 0.6068 28,629
L¼5 0.2965 0.5407 0.6326 0.0214 0.1835 29,674
CBO L ¼ 2 0.2129 0.4426 28,608
L¼5 0.5700 0.8618 0.5831 0.0777 0.3363 27,992
ECBO L ¼ 2 0.1765 0.9272 28,587
L¼5 0.4186 0.9776 0.7792 0.1007 0.0557 27,982
TWO L ¼ 2 0.1613 0.8465 28,579
L¼5 0.3306 0.8144 0.5654 0.0668 0.6810 27,977
Example 4 Sloan 1 2 157,457
King 0 1 157,103
PSO L¼2 0.0449 0.6963 157,095
L¼5 0.1106 0.9323 0.0624 0.0284 0.3706 160,705
CBO L ¼ 2 0.0229 0.9146 157,095
L¼5 0.9709 0.9856 0.1853 0.2102 0.3565 159,681
ECBO L ¼ 2 0.0620 0.9365 157,095
L¼5 0.5805 0.7778 0.2437 0.1277 0.1065 159,676
TWO L ¼ 2 0.0721 0.9800 157,095
L¼5 0.8206 0.8691 0.5953 0.4801 0.2781 159,675
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13.4 Numerical Examples 249
Table 13.3 Comparison of the results of different algorithms for wavefront reduction
Example Algorithm W1 W2 W3 W4 W5 Frms
Example 1 Sloan 1 2 20.1739
King 0 1 21.0798
PSO L¼2 0.3069 0.2202 20.1401
L¼5 0.3188 0.9852 0.7009 0.0489 0.118 17.2693
CBO L ¼ 2 0.8701 0.6144 20.1401
L¼5 0.1888 0.9093 0.8122 0.0210 0.8648 17.1544
ECBO L ¼ 2 0.8858 0.6517 20.1401
L¼5 0.1891 0.8720 0.9777 0.0199 0.4653 17.2239
TWO L ¼ 2 0.8711 0.6352 20.1401
L¼5 0.0101 0.9086 0.8164 0.0034 0.1742 17.1492
Example 2 Sloan 1 2 25.9092
King 0 1 26.6508
PSO L¼2 0.2053 0.3469 25.8411
L¼5 0.0879 0.3811 0.8933 0.012 0.1249 23.5891
CBO L ¼ 2 0.3181 0.5433 25.8438
L¼5 0.0713 0.6558 0.9556 0.0095 0.6861 23.4955
ECBO L ¼ 2 0.1855 0.3155 25.8411
L¼5 0.1056 0.8982 0.7698 0.0121 0.7255 23.4743
TWO L ¼ 2 0.5752 0.9624 25.8438
L¼5 0.0919 0.9571 0.6803 0.0130 0.0568 23.5090
Example 3 Sloan 1 2 18.3958
King 0 1 18.4698
PSO L¼2 0.0605 0.3289 18.3126
L¼5 0.0709 0.6589 0.9003 0.0677 0.4860 18.9964
CBO L ¼ 2 0.1382 0.8659 18.3235
L¼5 0.2174 0.4203 0.4590 0.0462 0.2162 19.2531
ECBO L ¼ 2 0.1688 0.8892 18.3232
L¼5 0.1883 0.7370 0.5928 0.0607 0.0945 18.6574
TWO L ¼ 2 0.0357 0.1982 18.3240
L¼5 0.0441 0.9101 0.7696 0.0139 0.6153 18.0211
Example 4 Sloan 1 2 32.3665
King 0 1 32.2875
PSO L¼2 0.0445 0.6963 32.2869
L¼5 0.2036 0.9340 0.0601 0.0151 0.5008 32.9486
CBO L ¼ 2 0.0361 0.8204 32.2869
L¼5 0.0469 0.9805 0.0890 0.0109 0.0406 32.7939
ECBO L ¼ 2 0.0145 0.6215 32.2869
L¼5 0.0665 0.9779 0.2937 0.0204 0.0726 32.8298
TWO L ¼ 2 0.0772 0.9898 32.2869
L¼5 0.3335 0.6943 0.6808 0.1610 0.0604 32.8845
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250 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
algorithms is tested on this model for bandwidth, profile, and wavefront optimiza-
tion problems. The results for these problems are given in Tables 13.1, 13.2, and
13.3, respectively. Quality of the results is provisioned in these tables.
This is the element clique graph of a rectangular FEM with four openings, as shown
in Fig. 13.4, having 760 nodes. The performance of the PSO, CBO, ECBO, and
TWO algorithms is tested on this model for bandwidth, profile, and wavefront
minimizing problems. The results for these problems are provided in Tables 13.1,
13.2, and 13.3, respectively.
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13.4 Numerical Examples 251
Fig. 13.4 The element clique graph of a rectangular FEM with four openings
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252 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
The FEM of an H-shaped shear wall with 4949 nodes is considered, as shown in
Fig. 13.6. The performance of the abovementioned algorithms is tested on this
model, and the results for bandwidth, profile, and wavefront minimizing problems
are given in Tables 13.1, 13.2, and 13.3, respectively.
13.5 Discussion
For Example 2, comparison of the results is shown in Figs. 13.7 and 13.8. The
convergence curves of the CBO, ECBO, PSO, and TWO algorithms are illustrated
in Figs. 13.9, 13.10, 13.11, and 13.12. The convergence histories show that these
four algorithms act in relatively the same way. To indicate the difference of the
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13.5 Discussion 253
18000
17500 L=5
L=2
17000
16500
16000
CBO ECBO TWO PSO Sloan King
23 L=2
22
21
CBO ECBO TWO PSO Sloan King
4
Fig. 13.9 Convergence x 10
curves of Example 2 for the 1.87
CBO, ECBO, PSO, and
TWO algorithms [1] 1.868
CBO
ECBO
profile - 2 Vectors
1.866 PSO
TWO
1.864
1.862
1.86
1.858
0 5 10 15 20 25
Iteration
convergence curves better, only 25 iterations have been shown. As can be seen from
Figs. 13.9, 13.10, 13.11, and 13.12, the CBO, ECBO, and TWO algorithms have
better convergence, search better the space of the problem, and obtain better results
compared to the PSO method.
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254 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
4
Fig. 13.10 The x 10
convergence history of 3
CBO
Example 2 for the CBO,
2.8 ECBO
ECBO, PSO, and TWO
PSO
algorithms [1]
2.6 TWO
Profile - 5 Vectors
2.4
2.2
1.8
0 5 10 15 20 25
Iteration
26.1 TWO
26.05
26
25.95
25.9
25.85
25.8
0 5 10 15 20 25
Iteration
TWO
80
60
40
20
0 5 10 15 20 25
Iteration
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References 255
The main purpose of this chapter was to show the performance and robustness of
the CBO, ECBO, and TWO for bandwidth, profile, and wavefront reduction of
matrices. From Table 13.1, it can be observed that the attained results from these
three algorithms are quite satisfactory compared to the well-known graph theoret-
ical method, four-step algorithm. CBO, its enhanced version, and TWO improve
the bandwidth values previously obtained by CSS and ACO algorithms, and these
values are the best results so far.
In profile and wavefront minimizing problems with L ¼ 2 and 5 methods, the aim
was to show the applicability of using different priority functions employing CBO,
ECBO, and TWO algorithms. Optimal coefficients for these functions are obtained by
optimization process, for decreasing the profile and wavefront of the stiffness matrices
of finite element models. From Tables 13.2 and 13.3, it can be observed that Sloan and
King’s methods can be improved in most cases using some new parameters and
coefficients. The weights achieved for different examples show that in the
two-parameter approach (L ¼ 2), more suitable profile and wavefront values can be
obtained than those of the Sloan and King’s algorithms. In the five-parameter method
(L ¼ 5), smaller profile and wavefront values can be attained than two-parameter
approach and Sloan and King’s algorithms except Example 4 that profile and
wavefront values of Sloan and King’s methods are smaller than those of the five-
parameter approach. It should be noted that in the L ¼ 5 algorithm, the active degrees
are not updated as in Sloan’s method. Therefore, one should not always expect a better
result when five adjusted parameters are utilized in place of two free parameters. The
value of profile and wavefront reduction in L ¼ 5 method proportion to Sloan and
King’s algorithm is more than that in L ¼ 2 method because of utilizing more graph
properties. For example, comparison of profile and wavefront results for Example 2 is
represented in Figs. 13.7 and 13.8, respectively. Among five parameters, the impor-
tance of parameter C2 is the highest and parameter C4 has the smallest effect.
A recently developed metaheuristic algorithm, tug of war optimization, is
employed, and from Tables 13.2 and 13.3, it can be seen that this algorithm obtains
good results like CBO and ECBO and in some cases achieves better values and the
best results so far.
Though the methods of this chapter are used for nodal ordering in the stiffness
method, however, the application of the methods can easily be extended to cycle or
generalized cycles ordering to optimize the bandwidth of the flexibility matrices [3,4].
References
1. Kaveh A, Bijari Sh (2016) Bandwidth, profile and wavefront reduction using PSO, CBO,
ECBO and TWO algorithms. Iran J Sci Technol (published online)
2. Kaveh A (1974) Applications of topology and matroid theory to the analysis of structures. Ph.
D. thesis, Imperial College of Science and Technology, London University, UK
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256 13 Bandwidth, Profile, and Wavefront Optimization Using CBO, ECBO, and TWO. . .
3. Kaveh A (2004) Structural mechanics: graph and matrix methods, 3rd edn. Research Studies
Press, Somerset
4. Kaveh A (2006) Optimal structural analysis, 2nd edn. Wiley, Chichester
5. Papademetrious CH (1976) The NP-completeness of bandwidth minimization problem.
Comput J 16:177–192
6. Gibbs NE, Poole WG, Stockmeyer PK (1976) An algorithm for reducing the bandwidth and
profile of a sparse matrix. SIAM J Numer Anal 12:236–250
7. Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Pro-
ceedings of the 24th national conference ACM, Bradon System Press, New Jersey, pp 157–172
8. Bernardes JAB, Oliveira SLGD (2015) A systematic review of heuristics for profile reduction
of symmetric matrices. Proc Comput Sci 51:221–230
9. King IP (1970) An automatic reordering scheme for simultaneous equations derived from
network systems. Int J Numer Methods Eng 2:523–533
10. Kaveh A (2014) Advances in metaheuristic algorithms for optimal design of structures.
Springer International Publishing, Switzerland
11. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 139:18–27
12. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
13. Kaveh A, Zolghadr A (2016) A novel meta-heuristic algorithm: tug of war optimization. Int J
Optim Civil Eng 6:469–493
14. Kaveh A, Sharafi P (2009) Nodal ordering for bandwidth reduction using ant system algorithm.
Eng Comput 26(3):313–323
15. Kaveh A, Sharafi P (2012) Ordering for bandwidth and profile minimization problems via
Charged System Search algorithm. Iran J Sci Technol Trans Civil Eng 36(C1):39–52
16. Kaveh A, Bijari S (2015) Bandwidth optimization using CBO and ECBO. Asian J Civil Eng 16
(4):535–545
17. Sloan SW (1986) An algorithm for profile and wavefront reduction of sparse matrices. Int J
Numer Methods Eng 23:1693–1704
18. Kaveh A, Roosta GR (1997) Graph-theoretical methods for profile reduction. In: Mouchel
Centenary conference on innovation in civil and structural engineering, Cambridge, UK
19. Rahimi Bondarabady HA, Kaveh A (2004) Nodal ordering using graph theory and a genetic
algorithm. Finite Elem Anal Des 40(9–10):1271–1280
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Chapter 14
Optimal Analysis and Design of Large-Scale
Domes with Frequency Constraints
14.1 Introduction
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14.2 Formulation of the Optimization Problem 259
Find X ¼ ½x1 , x2 , x3 , . . . :, xn
to minimize PðXÞ ¼ f ðXÞ f penalty ðXÞ
subject to
ð14:1Þ
ωj ωj * for some natural frequencies j
ωk ωk * for some natural frequencies k
ximin xi ximax
where X is the vector of the design variables, i.e., cross-sectional areas; n is the
number of optimization variables which depends on the element grouping scheme;
f(X) is the cost function, which is taken as the weight of the structure in a weight
optimization problem; and fpenalty(X) is the penalty function, which is used to make
the problem unconstrained. When some constraints are violated in a particular
solution, the penalty function magnifies the weight of the solution by taking values
bigger than one; P(X) is the penalized cost function or the objective function to be
minimized; ωj is the jth natural frequency of the structure with the corresponding
upper bound ωj*, while ωk is the kth natural frequency of the structure with the
corresponding lower bound ωk*; and ximin and ximax are the lower and upper bounds
for the design variable xi, respectively.
The cost function can be expressed as
X
nm
f ðX Þ ¼ ρ i L i Ai ð14:2Þ
i¼1
where nm is the number of structural members and ρi, Li, and Ai are the material
density, length, and cross-sectional area of the ith element.
The penalty function is defined as
X
q
f penalty ðXÞ ¼ ð1 þ ε1 :vÞε2 , v ¼ vi ð14:3Þ
i¼1
where q is the number of frequency constraints. The values for vi can be considered
as
(
0 if the ith constraint is satisfied
vi ¼ 1 ωi else ð14:4Þ
ω*
i
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260 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
in which K is the elastic stiffness matrix and M is the mass matrix of the
structure, ϕ i is the ith eigenvector (mode shape) corresponding to the ith
eigenvalue γ i, and the ith period (Ti) and circular frequency (ω i) are related to
the ith eigenvalue by
γ i ¼ ωi 2 ¼ ð2π=T i Þ2 ð14:6Þ
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14.3 Free Vibration Analysis of Structures 261
where E is the modulus of elasticity and Aij and Lij are the cross-sectional area and
the length of the element, respectively. In the submatrix dij, lij, mij, and nij are the
direction cosines of the element with respect to x-, y-, and z-axes, respectively:
xi xi yi yi zi zi
lij ¼ , mij ¼ , nij ¼ ð14:8Þ
Lij Lij Lij
It is apparent from Eq. (14.6) that the element stiffness matrix in Cartesian
coordinates is not invariant under rotation about any axis. Therefore, the global
stiffness matrix of a cyclically repetitive structure does not generally exhibit any
favorable pattern in Cartesian coordinates.
In order to use the desirable patterns of the global stiffness matrices of cyclically
symmetric structures, the element global stiffness matrix should be developed in a
cylindrical coordinate system. In such a coordinate system, the element stiffness
matrix is invariant under rotation about an axis of revolution. Thus, the global
stiffness matrix of a cyclically repeated structure exhibits a special pattern which is
highly desired for efficient eigensolutions. A three-dimensional truss element
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262 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
in which we have
xi x0 yi y0 xj x0 yj y0
loi ¼ , moi ¼ , loj ¼ , moj ¼ ð14:12Þ
r oi r oi r oj r oj
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r oi ¼ x2i þ y2i , r oj ¼ x2j þ y2j ð14:13Þ
The expanded form of the element global stiffness matrix in cylindrical coordi-
nates can then be derived as
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14.4 Efficient Eigensolution 263
2 3
s21 s1 s2 s1 nij s1 s3 s1 s4 s1 nij
6 s22 s2 nij s2 s3 s2 s4 s2 nij 7
6 7
EAij 6
6 n2ij s3 nij s4 nij n2ij 77
K rzθ ¼ ð14:14Þ
ij
Lij 6
6 s23 s3 s4 s3 nij 7
7
4 sym s24 s4 nij 5
n2ij
where
As it can be seen, this form of element stiffness matrix is invariant under rotation
about the axis of revolution. Therefore, all similar substructures have the same
stiffness matrix regardless of their rotational positions. Hence, the global stiffness
matrix of the structure embodies some interesting patterns, which can be used for
efficient eigensolution of the structure.
In relation to mass matrix, it should be noted that both lumped and consistent
mass matrices are invariant under rotation and therefore no transformation is
needed. Since additional lumped masses are added to the free nodes, the difference
between consistent and lumped mass matrices is negligible. A lumped mass matrix,
which lumps the masses of the elements in their end nodes, is utilized in this
chapter. Therefore, the mass matrix is a diagonal one.
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264 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
2 3
A B Bt
6 Bt A B 7
6 7
6 : : : 7
6 7
6 : : : 7 ð14:16Þ
6 7
6 : : : 7
6 7
4 Bt A B5
B Bt A
where subscripts K and M for A, B, and Bt refer to stiffness and mass matrices,
respectively, I is an n n identity matrix, and H is an n n unsymmetric permu-
tation matrix as
2 3
0 1 0
60 0 1 7
6 7
6 : : : 7
6 7
H¼6
6 : : : 7
7 ð14:19Þ
6 : : : 7
6 7
4 0 0 15
1 0 0
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14.5 Numerical Examples 265
where “det” stands for determinant. Here, the goal is to block diagonalize Ωi and
hence to decompose the main problem into some simpler subproblems. Let us
consider the following definitions:
A ¼ AK γ i AM
B ¼ BK γ i BM ð14:22Þ
Bt ¼ BKt γ i BMt
Combining Eqs. (14.17 and 14.18) with the above equations, Ωi can be written as
Ωi ¼ I A þ H B þ H t Bt ð14:23Þ
This form of Ωi can now be block diagonalized, and its jth block is as follows:
Ωij ¼ A þ λj B þ λj Bt ð14:24Þ
where λj is the jth eigenvalue of matrix H and the bar sign means conjugation of a
general complex number. Thus, the following equation holds
Y
n
detðΩi Þ ¼ det Ωij ð14:25Þ
j¼1
K j xi ¼ γ i M j xi , j ¼ 1, 2, 3, . . . , n ð14:26Þ
in which
K j ¼ AK þ λj BK þ λj BKt
ð14:27Þ
Mj ¼ AM þ λj BM þ λj BMt
In this section three numerical examples are studied in order to examine the
viability and efficiency of the proposed method. Democratic particle swarm opti-
mization (DPSO) as introduced by Kaveh and Zolghadr [14] is utilized as the
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266 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
The first test problem is the 600-bar single-layer dome structure shown in Fig. 14.3.
The entire structure is composed of 216 nodes and 600 elements generated by cyclic
repetition of a substructure having 9 nodes and 25 elements. The angle of cyclic
symmetry between similar substructures is 15 . A nonstructural mass of 100 kg is
attached to all free nodes. Table 14.1 summarizes the material properties, variable
bounds, and frequency constraints for this example. Figure 14.4 shows a substruc-
ture in more detail for nodal numbering and coordinates. Each of the elements of
this substructure is considered as a design variable. Thus, this is a size optimization
problem with 25 variables.
Using the classical method, it takes 2.6150 s to perform a typical analysis for this
structure, while the efficient method needs 0.0198 s, i.e., the efficient method is
about 132 times faster on a single analysis. Two different optimization cases are
performed on this example as well as the other two. In Case 1, the initial
eigenproblem is solved directly using MATLAB internal eigenvalue function;
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14.5 Numerical Examples 267
Table 14.1 Material properties, variable bounds, and frequency constraints for the 600-bar
single-layer dome
Property/unit Value
E (modulus of elasticity)/N/m2 2 1011
ρ (material density)/kg/m3 7850
Added mass/kg 100
Design variable lower bound/m2 1 104
Design variable upper bound/m2 100 104
Constraints on the first three frequencies/Hz ω1 5, ω3 7
this is called the classical method. In Case 2, the abovementioned efficient method
is used for the analysis part, i.e., the initial eigenproblem is decomposed into several
smaller ones, and then each of the subproblems is solved using the same MATLAB
function. In this example, 30 particles and 300 iterations (9000 analyses) are used
for both cases. The required computational time to complete a single optimization
run for Cases 1 and 2 is 27,326.25 s and 190.77 s, respectively. This means that the
optimization procedure could be performed about 143 times faster using the
efficient analysis method under the same circumstances. This example was solved
10 times using the efficient analysis method and the best result is presented in
Table 14.2.
The total computational time to perform ten optimization runs using the efficient
method is 1906.68 s (less than an hour), while it would have taken approximately
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268 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
Table 14.2 Optimized design for the 600-bar dome truss problem (added masses are not
included)
Cross-sectional area Cross-sectional area
Element no. (nodes) (cm2) Element no. (nodes) (cm2)
1 (1–2) 1.365 14 (5–13) 5.529
2 (1–3) 1.391 15 (5–14) 7.007
3 (1–10) 5.686 16 (6–7) 5.462
4 (1–11) 1.511 17 (6–14) 3.853
5 (2–3) 17.711 18 (6–15) 7.432
6 (2–11) 36.266 19 (7–8) 4.261
7 (3–4) 13.263 20 (7–15) 2.253
8 (3–11) 16.919 21 (7–16) 4.337
9 (3–12) 13.333 22 (8–9) 4.028
10 (4–5) 9.534 23 (8–16) 1.954
11 (4–12) 9.884 24 (8–17) 4.709
12 (4–13) 9.547 25 (9–17) 1.410
13 (5–6) 7.866 Weight (kg) 6344.55
273,112.84 s (more than 3 days) to perform the same runs using the classical
method. Table 14.3 presents the first five natural frequencies of the optimized
structure. It can be seen that the constraints are fully satisfied. These frequencies
are in full agreement with the results of the classical analysis method up to ten
significant digits. The mean weight of the structures found in ten runs is 6674.71 kg
with a standard deviation of 473.21 kg. Figure 14.5 shows the convergence curve of
the best result for the 600-bar dome truss using the efficient method.
The second test problem solved in this study was the weight minimization of the
1180-bar dome truss structure shown in Fig. 14.6. The entire structure is composed
of 400 nodes and 1180 elements generated by cyclic repetition of a substructure
with 20 nodes and 59 elements. The angle of cyclic symmetry between similar
substructures is 18 . A nonstructural mass of 100 kg is attached to all free nodes.
Table 14.4 summarizes the material properties, variable bounds, and frequency
constraints for this example. Figure 14.7 shows a substructure in more detail for
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14.5 Numerical Examples 269
Fig. 14.5 Convergence curve of the best result for the 600-bar dome truss using the efficient
method [1]
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270 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
Table 14.4 Material properties, variable bounds, and frequency constraints for the 1180-bar
dome truss
Property/unit Value
E (modulus of elasticity)/N/m2 2 1011
ρ (material density)/kg/m3 7850
Added mass/kg 100
Design variable lower bound/m2 1 104
Design variable upper bound/m2 100 104
Constraints on the first three frequencies/Hz ω1 7, ω3 9
nodal numbering. Table 14.5 summarizes the coordinates of the nodes in Cartesian
coordinate system. Each of the elements of this substructure is considered as a
design variable. Thus, this is a size optimization problem with 59 variables.
A single analysis takes up to 11.3575 s of computational time using the classical
method. The required computational time for a similar analysis using the efficient
method is only 0.0720 s. This means that the efficient method is about 157 times
faster for a single analysis. About 100 particles and 500 iterations (50,000 analyses)
are used for optimization of this test problem. The required computational time to
complete a single run for Case 2 is 7095.56 s. Figure 14.8 shows the variation of the
computational time with the number of analyses for Case 1. According to the figure,
it is estimated that it would take 800,160 s to perform the same optimization run for
Case 1 (50,000 analyses). Therefore, the optimization procedure could be
performed about 113 times faster under the same circumstances using the efficient
analysis method. Again, this example was solved ten times using the efficient
analysis method and the best result is presented in Table 14.6.
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14.5 Numerical Examples 271
Table 14.5 Coordinates of the nodes of the main structure (the 1180-bar dome truss)
Node no. Coordinates (x, y, z) Node no. Coordinates (x, y, z)
1 (3.1181, 0.0, 14.6723) 11 (4.5788, 0.7252, 14.2657)
2 (6.1013, 0.0, 13.7031) 12 (7.4077, 1.1733, 12.9904)
3 (8.8166, 0.0, 12.1354) 13 (9.9130, 1.5701, 11.1476)
4 (11.1476, 0.0, 10.0365) 14 (11.9860, 1.8984, 8.8165)
5 (12.9904, 0.0, 7.5000) 15 (13.5344, 2.1436, 6.1013)
6 (14.2657, 0.0, 4.6358) 16 (14.4917, 2.2953, 3.1180)
7 (14.9179, 0.0, 1.5676) 17 (14.8153, 2.3465, 0.0)
8 (14.9179, 0.0, 1.5677) 18 (14.4917, 2.2953, 3.1181)
9 (14.2656, 0.0, 4.6359) 19 (13.5343, 2.1436, 6.1014)
10 (12.9903, 0.0, 7.5001) 20 (3.1181, 0.0, 13.7031)
Fig. 14.8 Variation of the computational time with a number of analyses for Case 1 (1180-bar
dome truss)
It takes 68,933.06 s to perform ten optimization runs using the efficient method
for this example, while it would have taken approximately 7,773,580 s (about
90 days) to perform the same runs using the classical method. Table 14.7 presents
the first five natural frequencies of the optimized structure for this example. The
mean weight of the structures found in ten runs is 38,294.45 kg with a standard
deviation of 550.5 kg. Figure 14.9 shows the convergence curve of the best result
for the 1180-bar dome truss using the efficient method.
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272 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
Table 14.6 Optimized design for the 1180-bar dome truss problem (added masses are not
included)
Cross-sectional area Cross-sectional area
Element no. (nodes) (cm2) Element no. (nodes) (cm2)
1 (1–2) 7.926 31 (8–9) 34.642
2 (1–11) 10.426 32 (8–17) 19.860
3 (1–20) 2.115 33 (8–18) 25.079
4 (1–21) 14.287 34 (8–28) 18.965
5 (1–40) 3.846 35 (9–10) 47.514
6 (2–3) 5.921 36 (9–18) 28.133
7 (2–11) 7.955 37 (9–19) 33.023
8 (2–12) 6.697 38 (9–29) 32.263
9 (2–20) 1.889 39 (10–19) 33.401
10 (2–22) 11.881 40 (10–30) 1.344
11 (3–4) 7.121 41 (11–21) 9.327
12 (3–12) 6.080 42 (11–22) 7.202
13 (3–13) 6.599 43 (12–22) 6.792
14 (3–23) 7.772 44 (12–23) 6.228
15 (4–5) 9.358 45 (13–23) 6.601
16 (4–13) 6.213 46 (13–24) 6.584
17 (4–14) 8.200 47 (14–24) 8.320
18 (4–24) 7.799 48 (14–25) 8.844
19 (5–6) 11.752 49 (15–25) 11.254
20 (5–14) 7.494 50 (15–26) 12.162
21 (5–15) 9.696 51 (16–26) 13.854
22 (5–25) 9.177 52 (16–27) 13.844
23 (6–7) 17.326 53 (17–27) 17.536
24 (6–15) 11.797 54 (17–28) 20.551
25 (6–16) 14.002 55 (18–28) 24.072
26 (6–26) 11.562 56 (18–29) 27.287
27 (7–8) 23.981 57 (19–29) 32.965
28 (7–16) 12.996 58 (19–30) 36.940
29 (7–17) 16.591 59 (20–40) 3.837
30 (7–27) 15.910 Weight (kg) 37,779.81
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14.5 Numerical Examples 273
Fig. 14.9 Convergence curve of the best result for the 1180-bar dome truss using the efficient
method [1]
The third test problem solved in this chapter was the weight minimization of the
1410-bar double-layer dome truss as shown in Fig. 14.10. The entire structure is
composed of 390 nodes and 1410 elements generated by cyclic repetition of a
substructure with 13 nodes and 47 elements. The angle of cyclic symmetry between
similar substructures is 12 . A nonstructural mass of 100 kg is attached to all free
nodes. Table 14.8 summarizes the material properties, variable bounds, and fre-
quency constraints for this example. Figure 14.11 shows a substructure in more
detail for nodal numbering. Table 14.9 presents the coordinates of the nodes in
Cartesian coordinate system. Each of the elements of this substructure is considered
as a design variable. Thus, this is a size optimization problem with 47 variables.
Required computational times for classical and efficient methods are 11.7101
and 0.0140 s, respectively. Like the previous example, 100 particles and 500 itera-
tions (50,000 analyses) are used for optimization of this test problem. The required
computational time to complete a single run for Case 2 is 3871.62 s. Figure 14.12
shows the variation of the computational time with the number of analyses for Case
1. According to the figure, it is estimated that it would take 950,240 s to perform the
same optimization run for Case 1 (50,000 analyses). Therefore, the optimization
procedure could be performed about 245 times faster under the same circumstances
using the efficient analysis method. This example was solved ten times using the
efficient analysis method and the best result is presented in Table 14.10.
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274 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
Table 14.8 Material properties, variable bounds, and frequency constraints for the 1410-bar
dome truss
Property/unit Value
E (modulus of elasticity)/N/m2 2 1011
ρ (material density)/kg/m3 7850
Added mass/kg 100
Design variable lower bound/m2 1 104
Design variable upper bound/m2 100 104
Constraints on the first three frequencies/Hz ω1 7, ω3 9
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14.5 Numerical Examples 275
Table 14.9 Coordinates of the nodes of the main substructure (the 1410-bar dome truss)
Node no. Coordinates (x, y, z) Node no. Coordinates (x, y, z)
1 (1.0, 0.0, 4.0) 8 (1.989, 0.209, 3.0)
2 (3.0, 0.0, 3.75) 9 (3.978, 0.418, 2.75)
3 (5.0, 0.0, 3.25) 10 (5.967, 0.627, 2.25)
4 (7.0, 0.0, 2.75) 11 (7.956, 0.836, 1.75)
5 (9.0, 0.0, 2.0) 12 (9.945, 1.0453, 1.0)
6 (11.0, 0.0, 1.25) 13 (11.934, 1.2543, 0.5)
7 (13.0, 0.0, 0.0)
Fig. 14.12 Variation of the computational time with a number of analyses for Case 1 (1410-bar
dome truss)
It takes 38,310.43 s to perform ten optimization runs using the efficient method
for this example, while it would have taken approximately 9,386,055 s (about
108 days) to perform the same runs using the classical method. Table 14.11 presents
the first five natural frequencies of the optimized structure for this example. The
mean weight of the structures found in ten runs is 38,294.45 kg with a standard
deviation of 550.5 kg. Figure 14.13 shows the convergence curve of the best result
for the 1410-bar dome truss using the efficient method.
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276 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
Table 14.10 Optimized design for the 1410-bar dome truss problem (added masses are not
included)
Element Element
no. (nodes) Cross-sectional area (cm2) no. (nodes) Cross-sectional area (cm2)
1 (1–2) 7.209 25 (8–9) 2.115
2 (1–8) 5.006 26 (8–14) 4.923
3 (1–14) 38.446 27 (8–15) 4.047
4 (2–3) 9.438 28 (8–21) 5.906
5 (2–8) 4.313 29 (9–10) 3.392
6 (2–9) 1.494 30 (9–15) 1.902
7 (2–15) 8.455 31 (9–16) 4.381
8 (3–4) 9.488 32 (9–22) 8.442
9 (3–9) 3.480 33 (10–11) 5.011
10 (3–10) 3.495 34 (10–16) 3.577
11 (3–16) 16.037 35 (10–17) 2.805
12 (4–5) 9.796 36 (10–23) 2.024
13 (4–10) 2.413 37 (11–12) 6.709
14 (4–11) 5.681 38 (11–17) 5.054
15 (4–17) 15.806 39 (11–18) 3.259
16 (5–6) 8.078 40 (11–24) 1.063
17 (5–11) 3.931 41 (12–13) 5.934
18 (5–12) 6.099 42 (12–18) 7.057
19 (5–18) 10.771 43 (12–19) 5.745
20 (6–7) 13.775 44 (12–25) 1.185
21 (6–12) 4.231 45 (13–19) 7.274
22 (6–13) 6.995 46 (13–20) 4.798
23 (6–19) 1.837 47 (13–26) 1.515
24 (7–13) 4.397 Weight (kg) 10,453.84
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14.6 Concluding Remarks 277
Fig. 14.13 Convergence curve of the best result for the 1410-bar dome truss using the efficient
method [1]
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278 14 Optimal Analysis and Design of Large-Scale Domes with Frequency Constraints
References
1. Kaveh A, Zolghadr A (2016) Optimal analysis and design of large-scale domes with frequency
constraints. Smart Struct Syst 18(5):733–754
2. Grandhi RV (1993) Structural optimization with frequency constraints—a review. AIAA J 31
(12):2296–2303
3. Taylor JE (1967) Minimum mass bar for axial vibration at specified natural frequency. AIAA J
5(10):1911–1913
4. Armand JL (1971) Minimum mass design of a plate like structure for specified fundamental
frequency. AIAA J 9(9):1739–1745
5. Cardou A, Warner WH (1974) Minimum mass design of sandwich structures with frequency
and section constraints. J Optim Theory Appl 14(6):633–647
6. Elwany MHS, Barr ADS (1979) Minimum weight design of beams in torsional vibration with
several frequency constraints. J Sound Vib 62(3):411–425
7. Lin JH, Chen WY, Yu YS (1982) Structural optimization on geometrical configuration and
element sizing with static and dynamic constraints. Comput Struct 15(5):507–515
8. Konzelman CJ (1986) Dual methods and approximation concepts for structural optimization.
MSc thesis, Department of Mechanical Engineering, University of Toronto, Canada
9. Grandhi RV, Venkayya VB (1988) Structural optimization with frequency constraints. AIAA J
26(7):858–866
10. Sedaghati R, Suleman A, Tabarrok B (2002) Structural optimization with frequency con-
straints using finite element force method. AIAA J 40(2):382–388
11. Lingyun W, Mei Z, Guangming W, Guang M (2005) Truss optimization on shape and sizing
with frequency constraints based on genetic algorithm. J Comput Mech 35(5):361–368
12. Gomes MH (2011) Truss optimization with dynamic constraints using a particle swarm
algorithm. Expert Syst Appl 38(1):957–968
13. Kaveh A, Zolghadr A (2014) Comparison of nine meta-heuristic algorithms for optimal design
of truss structures with frequency constraints. Adv Eng Softw 76:9–30
14. Kaveh A, Zolghadr A (2014) Democratic PSO for truss layout and size optimization with
frequency constraints. Comput Struct 130:10–21
15. Kaveh A, Rahami H (2006) Block diagonalization of adjacency and Laplacian matrices for
graph product; applications in structural mechanics. Int J Numer Methods Eng 68(1):33–63
16. Kaveh A, Rahami H (2007) Compound matrix block diagonalization for efficient solution of
eigenproblems in structural mechanics. Acta Mech 188(3–4):155–166
17. Kaveh A (2013) Optimal analysis of structures by concepts of symmetry and regularity.
Springer, Wien, New York, NY
18. Koohestani K, Kaveh A (2010) Efficient buckling and free vibration analysis of cyclically
repeated space truss structures. Finite Elem Anal Des 46(10):943–948
19. Courant R (1943) Variational methods for the solution of problems of equilibrium and
vibrations. Bull Am Math Soc 49:1–23
20. Leung AYT (1980) Dynamic analysis of periodic structures. J Sound Vib 72(4):451–467
21. Williams FW (1986) An algorithm for exact eigenvalue calculations for rotationally periodic
structures. Int J Numer Methods Eng 23(4):609–622
22. Vakakis AF (1992) Dynamics of a nonlinear periodic structure with cyclic symmetry. Acta
Mech 95(1–4):197–226
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References 279
23. Karpov EG, Stephen NG, Dorofeev DL (2002) On static analysis of finite repetitive structures
by discrete Fourier transform. Int J Solids Struct 39(16):4291–4310
24. Liu L, Yang H (2007) A paralleled element-free Galerkin analysis for structures with cyclic
symmetry. Eng Comput 23(2):137–144
25. El-Raheb M (2011) Modal properties of a cyclic symmetric hexagon lattice. Comput Struct
89:2249–2260
26. Zingoni A (2014) Group-theoretic insights on the vibration of symmetric structures in engi-
neering. Philos Trans R Soc A 372:20120037
27. Tran D-M (2014) Reduced models of multi-stage cyclic structures using cyclic symmetry
reduction and component mode synthesis. J Sound Vib 333:5443–5463
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Chapter 15
Optimum Design of Large-Scale Truss
Towers Using Cascade Optimization
15.1 Introduction
High number of design variables, large size of the search space, and control of a
great number of design constraints are major preventive factors in performing
optimum design of real-world structures in a reasonable time. This chapter presents
an accurate and efficient technique for optimal design of truss towers with large
number of design variables to illustrate its applicability to optimum design of
practical structures [1].
Cascade sizing optimization utilizing a series of design variable configurations
(DVCs) is used in this study. Several DVCs are constructed in order to utilize a
different configuration at each cascade optimization stage. Each new cascade stage
is coupled with the previous one by initializing the new stage using the finally
attained optimum design of the previous stage. The first stages of the cascade
procedure are executed with the coarsest DVCs, and the final cascade stages utilize
the finest DVCs in order to handle large numbers of design variables. In all stages of
the procedure, enhanced colliding bodies optimization (ECBO) is employed. The
multi-DVC cascade optimization performs better than non-cascade procedure in all
the considered examples. High solution accuracy and convergence speed of the
proposed method are shown through three test examples [1].
In the last decades, a number of optimization techniques have been developed
and used for structural optimization problems. The aim of the optimization is to
minimize an objective function that is often considered as the cost of the structure
or a quantity directly proportional to the cost under certain constraints. These
constraints may consist of any engineering demand parameter like stresses, dis-
placements, maximum inter-story drift, etc. Recent years have witnessed an
increasing interest in the development and application of metaheuristic algorithms
that are effective and robust techniques for optimization problems. These algo-
rithms are often population based, and they search for the global optimum of the
problem through sharing information to cooperate and/or compete among the
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15.2 Cascade Sizing Optimization Utilizing Series of Design Variable Configurations 283
No single optimizer can successfully solve all the structural design problems.
Cascade optimization strategy was proposed to alleviate this deficiency which
utilizes several optimizers, one followed by another in a specified sequence, to
solve a problem [8]. In the first stage of the cascade procedure, the first optimizer
starts from a user-specified design, known as the “cold-start.” The intermediate
optimal solution reached in the first cascade stage, which may be perturbed using a
pseudo-random technique, is called a “hot-start” and is used to initiate the second
optimization stage. Accordingly, each optimization stage of the cascade procedure
starts from the optimum design achieved in the previous stage (possibly perturbed).
Thus, each cascade stage initiates from a hot-start and produces a new hot-start for
the next stage. This way the autonomous computations of successive optimization
stages are coupled. In general, the optimization algorithm implemented at each
stage of a cascade process may or may not be the same. Cascade optimization has
been implemented using different deterministic and/or probabilistic optimizers in
the cascade stages (Charmpis et al. [3]).
A series of appropriate DVCs for the sizing optimization problem under consider-
ation is formed, in order to utilize a different configuration at each cascade
optimization stage. Each new cascade stage is coupled with the previous one by
initializing the new stage using the finally attained optimum design of the previous
one (Charmpis et al. [4]). The first stages of the cascade procedure are executed
with the coarsest DVCs aiming at a basic non-detailed search of the full design
space. This search is facilitated by the manageable DVCs handled to avoid confus-
ing the optimizer with huge design spaces. Thus, the areas of appropriate design
variable values are identified by detecting near optimum solutions among the
relatively limited design options provided. As the numbers of design variables
processed in the cascade stages become larger, more detailed representation of
the full design space is offered and the optimizer is given the opportunity to
improve the quality of the optimal solution reached. In the final cascade stages
utilizing the finest DVCs, relatively small adjustments to an already good-quality
design occur, in an effort to identify (or at least approach) the globally optimum
design. Hence, the early optimization stages of the cascade procedure serve the
purpose of basic design space exploration, while the later stages aim at fine-tuning
of the achieved optimal solution (Charmpis et al. [4]).
This multi-DVC cascade computational procedure can be implemented using an
arbitrary optimization algorithm. In this study, ECBO (Kaveh and Ilchi Ghazaan
[7]) that is presented in the next section is utilized in all stages. Flowchart of the
Multi-DVC cascade optimization procedure is shown in Fig. 15.1.
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284 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
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15.3 Enhanced Colliding Bodies Optimization 285
Step 1: Initialization
Each solution candidate xi is considered as a colliding body (CB) and the initial
positions of all CBs are determined randomly in an m-dimensional search space.
where xi0 is the initial solution vector of the ith CB. Here, xmin and xmax are the
bounds of design variables; rand is a random vector in which each component is
in the interval [0, 1]; the sign “ ” denotes an element-by-element multiplication;
n is the number of CBs.
Step 2: Defining mass
Each CB has a specified mass defined as
1
fitðkÞ
mk ¼ 1
, k ¼ 1, 2, . . . , n ð15:2Þ
Xn 1
i¼1 fitðiÞ
where fit(i) represents the objective function value of the ith CB.
Step 3: Saving
Colliding memory (CM) is utilized to save a number of the best-so-far solutions.
In this study, the size of the CM is taken as n/10. At each iteration, solution
vectors saved in CM are added to the population, and the same number of current
worst CBs are deleted. Finally, CBs are sorted according to their masses in a
decreasing order.
Step 4: Creating groups
In order to select pairs of objects for collision, CBs are divided into two equal
groups: (i) stationary group and (ii) moving group. Moving objects collide with
stationary objects to improve their positions and push stationary objects toward
better positions.
Step 5: Criteria before the collision
The velocity of the stationary bodies before collision is zero so
n
vi ¼ 0, i ¼ 1, 2, . . . , ð15:3Þ
2
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286 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
miþn2 þ εmiþn2 viþn2 n
v0i ¼ i ¼ 1, 2, . . . , ð15:5Þ
mi þ miþn2 2
0
The velocity of each moving CB after the collision (vi ) is defined by
mi εmin2 vi n n
v0i ¼ i¼ þ 1 , þ 2 , ... , n ð15:6Þ
mi þ m in2 2 2
ε is the coefficient of restitution (COR) that decreases linearly from unit to zero
iter
ε¼1 ð15:7Þ
iter max
where iter is the current iteration number and itermax is the total number of
iterations for optimization process.
Step 7: Updating CBs
New positions of CBs are updated according to their velocities after the collision
and the positions of stationary CBs. Therefore, the new position of each station-
ary CB is
0 n
xi new ¼ xi þ rand∘vi , i ¼ 1, 2, . . . , ð15:8Þ
2
where xij is the jth variable of the ith CB. xj,min and xj,max, respectively, are the
lower and upper bounds of the jth variable. rnd is a random number in the
interval [0,1]. In this study, pro is set to 0.25.
Step 9: Termination condition check
After the predefined maximum evaluation number, the optimization process is
terminated.
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15.4 Design Examples 287
Three large-scale truss structures are optimized for minimum volume with the
cross-sectional areas of the members being the design variables to verify the
efficiency of the multi-DVC cascade optimization. A population of 20 CBs is
used for the first and second examples and 30 CBs are utilized for the last problem.
The optimization process in each stage except the last one is terminated after a fixed
number of iterations without any improvement. This value is considered as the
minimum of the number of design variables in the stage as 30. When the total
number of iterations is equal to 1000, the process is terminated. In all problems, the
CBs are allowed to select discrete values from the permissible list of cross sections
(real numbers are rounded to the nearest integer in the each iteration). The well-
known penalty approach is employed to handle the constraints (Kaveh and Ilchi
Ghazaan [7]). The algorithms are coded in MATLAB, and the structures are
analyzed using the direct stiffness method.
σþ
i ¼ 0:6Fy ð15:11Þ
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288 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
8
>
> λ2i 5 3λi λ3i
>
< 1 F y = þ for λi < C
2C2c 3 8Cc 8C3c
σ ¼ ð15:12Þ
i
> 2
> 12π E
>
: for λi Cc
23λ2i
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15.4 Design Examples 289
Table 15.1 Design variable configurations utilized for the 582-bar tower problem
Number of design Design variables in the group (design variable
variables in stages configurations)
Stage 1 8 [1 6 9]; [2 4 7 10]; [3 5 8 11]; [12 13 14]; [19 22 25 28 31];
[32]; [15 17 20 23 26 29]; [16 18 21 24 27 30]
Stage 2 15 [1 6 9]; [2 4]; [7 10]; [3 5]; [8 11]; [12]; [13]; [14]; [19 22
25]; [28 31]; [32]; [15 17 20]; [23 26 29]; [16 18 21];
[24 27 30]
Fig. 15.3 Convergence curves of non-cascade (solid lines) and cascade optimization procedures
(dotted lines) obtained in the 582-bar tower problem [1]
optimization procedures are 8980 and 6620 analyses, respectively. It means that the
algorithm manages to find a near optimal solution in the early iterations while it
continues searching the search space until the last iterations. Convergence curves
are depicted in Fig. 15.3. The final volumes achieved in stage 1 (containing 8 design
variables) and stage 2 (containing 15 design variables) are 1,438,697 m3 and
1,363,348 m3, respectively. These stages are terminated in 95th and 197th itera-
tions. It can be seen that the convergence rate of the cascade optimization pro-
cedures is higher than the non-cascade procedure.
Figure 15.4 shows the schematic of a 942-bar tower truss. This example has been
analyzed by many researchers considering 59 design variables (Hasançebi [10]). In
this study, the design variables are increased to 76 and the performance constraints,
material properties, and other conditions are the same as those of the first example.
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290 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
Figure 15.5 shows the member groups. Three stages with 16, 28, and 76 design
variables are considered to solve this problem. The DVCs are shown in Table 15.2.
The design obtained by cascade optimization procedures is 3,323,028 m3, and the
best design attained without cascading is 3,376,968 m3. These values are found after
18,320 and 19,960 analyses, respectively. The proposed method can reach the best
design of non-cascade procedure after about 11,060 analyses. Convergence history
diagrams are depicted in Fig. 15.6. The final volume found in stage 1 (containing
16 design variables) in 112th iteration is 4,467,989 m3. Stage 2 (containing 28 design
variables) terminated in 287th iteration and its corresponding value is 3,809,870 m3.
It can be seen that the curve of the multi-DVC cascade optimization lies below those
of the non-cascading procedure.
The schematic of a 2386-bar tower truss is shown in Fig. 15.7 as the last design
example. This example is studied here for the first time. The Performance con-
straints, material properties, and other conditions are the same as those of the first
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15.4 Design Examples 291
1st, 2nd and 3rd stories 13th, 14th and 15th stories 21st, 22nd and 23rd stories
Table 15.2 Design variable configurations utilized for the 942-bar tower problem
Number of design Design variables in the group (design variable
variables in stages configurations)
Stage 1 16 [1]; [2–6]; [7–12]; [13–18]; [19–24]; [25–29]; [30–35];
[36]; [37–43]; [44–51]; [52–58]; [59]; [60–64]; [65–70];
[71–75]; [76]
Stage 2 28 [1]; [2 3]; [4–6]; [7–9]; [10–12]; [13–15]; [16–18]; [19–21];
[22–24]; [25 26]; [27–29]; [30–32]; [33–35]; [36]; [37–39];
[40–43]; [44–47]; [48–51]; [52–54]; [55–58]; [59]; [60 61];
[62–64]; [65–67]; [68–70]; [71 72]; [73–75]; [76]
Fig. 15.6 Convergence curves of non-cascade (solid lines) and cascade optimization procedures
(dotted lines) obtained in the 942-bar tower problem [1]
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292 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
example. The elements are divided into 220 groups and member groups are
presented in Fig. 15.8. Four stages are considered to optimize this example. The
number of design variables in stages 1, 2, 3, and 4 are 21, 42, 84, and 220, respec-
tively. Table 15.3 lists the DVCs.
The proposed method obtained 12,535,919 m3 after 29,010 analyses which is
better than 14,086,857 m3 found by the non-cascade procedure after 29,670 ana-
lyses. The best design of non-cascade procedure can be achieved by multi-DVC
cascade optimization after only 6900 analyses. Convergence curves are compared
in Fig. 15.9. The final volumes achieved in stage 1 (containing 21 design variables),
stage 2 (containing 42 design variables), and stage 3 (containing 84 design vari-
ables) are 14,504,868 m3, 13,416,104 m3, and 12,862,132 m3, respectively. These
stages are terminated in 225th, 501th, and 761th iterations. It can be seen from the
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15.4 Design Examples 293
Table 15.3 Design variable configurations utilized for the 2386-bar tower problem
Number of design Design variables in the group (design variable
variables in stages configurations)
Stage 1 21 [1–10]; [11–20]; [21–31]; [32–41]; [42–51]; [52–62];
[63–72]; [73–82]; [83–92] [93–103]; [104–113]; [114–124];
[125–135]; [136–146]; [147–156]; [157–167] [168–178];
[179–188]; [189–199]; [200–210]; [211–220]
Stage 2 42 [1:4]; [5:10]; [11:15]; [16:20]; [21:25]; [26:31]; [32:36];
[37:41]; [42:46]; [47:51]; [52:56]; [57:62]; [63:67]; [68:72];
[73:77]; [78:82]; [83:87]; [88:92]; [93:97]; [98:103];
[104:108]; [109:113]; [114:118]; [119:124]; [125:129];
[130:135]; [136:140]; [141:146]; [147:151]; [152:156];
[157:161]; [162:167]; [168:172]; [173:178]; [179:183];
[184:188]; [189:193]; [194:199]; [200:204]; [205:210];
[211:215]; [216:220]
Stage 3 84 [1]; [2–4]; [5–7]; [8–10]; [11 12]; [13–15]; [16 17]; [18–20];
[21 22]; [23–25]; [26–28]; [29–31]; [32 33]; [34–36];
[37 38]; [39–41]; [42 43]; [44–46]; [47 48]; [49–51];
[52 53]; [54–56]; [57–59]; [60–62]; [63 64]; [65–67];
[68 69]; [70–72]; [73 74]; [75–77]; [78 79]; [80–82];
[83 84]; [85–87]; [88 89]; [90–92]; [93 94]; [95–97];
[98–100]; [101–103]; [104 105]; [106–108]; [109 110];
[111–113]; [114 115]; [116–118]; [119–121]; [122–124];
[125 126]; [127–129]; [130–132]; [133–135]; [136 137];
[138–140]; [141–143]; [144–146]; [147 148]; [149–151];
[152 153]; [154–156]; [157 158]; [159–161]; [162–164];
[165–167]; [168 169]; [170–172]; [173–175]; [176–178];
[179 180]; [181–183]; [184 185]; [186–188]; [189 190];
[191–193]; [194–196]; [197–199]; [200 201]; [202–204];
[205–207]; [208–210]; [211 212]; [213–215]; [216 217];
[218–220]
Fig. 15.9 Convergence curves of non-cascade (solid lines) and cascade optimization procedures
(dotted lines) obtained in the 2386-bar tower problem [1]
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294 15 Optimum Design of Large-Scale Truss Towers Using Cascade Optimization
Fig. 15.10 Element stress ratio obtained in the 2386-bar tower problem: (a) non-cascade optimi-
zation procedures and (b) cascade optimization procedures
plots that the intermediate designs found by proposed method are always better than
those found by non-cascade procedure. The stress ratios for all the members are
shown in Fig. 15.10. The maximum values of the stress ratio for non-cascade and
cascade procedures are 97.57 % and 99.96 %, respectively.
Three numerical examples chosen from size optimum design of truss towers with
large number of design variables are studied to test and verify efficiency of the
multi-DVC cascade optimization that utilizes a different DVC in each stage of the
cascade optimization procedure, as well as to illustrate its applicability for optimum
design of practical structures. In the 32-variable design example, the best volumes
obtained by non-cascade and cascade optimization procedures were approximately
the same, but cascade optimization procedure had a better convergence rate. The
optimum volume found by cascade optimization procedure in the 76-variable
design example was about 2 % lighter than that obtained by non-cascade procedure.
Also, the required number of iterations for achieving the best design of non-cascade
procedure was also decreased 50 % by the proposed method. In the 220-variable
design example, the design obtained by the cascade optimization procedures is
about 11 % lighter than the best design attained without cascading. The required
number of iterations for achieving the best design of non-cascade procedure was
also decreased 50 % by the proposed method. It can be concluded that by increasing
the size of the search space, the differences between the accuracy of the cascading
and non-cascading procedures considerably increase. To sum up, multi-DVC
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References 295
References
1. Kaveh A, Ilchi Ghazaan M (2016) Optimum design of large-scale truss towers using cascade
optimization. Acta Mech 227(9):2645–2656
2. Kaveh A (2014) Advances in metaheuristic algorithms for optimal design of structures.
Springer, Switzerland
3. Charmpis DC, Lagaros ND, Papadrakakis M (2005) Multi-database exploration of large design
spaces in the framework of cascade evolutionary structural sizing optimization. Comput
Methods Appl Mech Eng 194:3315–3330
4. Charmpis DC, Lagaros ND, Papadrakakis M (2015) Cascade structural sizing optimization
with large numbers of design variables. Adv Eng Softw (submitted)
5. Kaveh A, Ilchi Ghazaan M (2014) Enhanced colliding bodies optimization for design problems
with continuous and discrete variables. Adv Eng Softw 77:66–75
6. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method.
Comput Struct 139:18–27
7. Kaveh A, Ilchi Ghazaan M (2015) A comparative study of CBO and ECBO for optimal design
of skeletal structures. Comput Struct 153:137–147
8. Patnaik SN, Coroneos RM, Hopkins DA (1997) A cascade optimization strategy for solution of
difficult design problems. Int J Numer Methods Eng 40:2257–2266
9. American Institute of Steel Construction (AISC) (1989) Manual of steel construction—allow-
able stress design, 9th edn. AISC, Chicago, IL
10. Hasançebi O (2008) Adaptive evolution strategies in structural optimization: enhancing their
computational performance with applications to large-scale structures. Comput Struct
86:119–132
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Chapter 16
Vibrating Particles System Algorithm
for Truss Optimization with Frequency
Constraints
16.1 Introduction
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16.2 Statement of the Optimization Problem 299
In this chapter, the objective is to minimize the weight of the structure while
satisfying some constraints on natural frequencies. Each variable should be chosen
within a permissible range. The mathematical formulation of these problems can be
expressed as follows:
Find fXg ¼ x1 ; x2 ; ::; xng
X
nm
to minimize W ð fX gÞ ¼ ρ i Ai L i
i¼1
8 ð16:1Þ
>
> ω j ω∗
< j
subjected to : ω k ω∗
>
>
k
:
xi min xi ximax
where {X} is the vector containing the design variables; ng is the number of design
variables; W({X}) presents the weight of the structure; nm is the number of
elements of the structure; ρi, Ai, and Li denote the material density, the cross-
sectional area, and the length of the ith member, respectively; ωj is the jth natural
frequency of the structure and ωj* is its upper bound; ωk is the kth natural frequency
of the structure and ωk* is its lower bound; xi min and xi max are the lower and upper
bounds of the design variable xi, respectively.
To handle the constraints, the well-known penalty approach is employed. Thus,
the objective function is redefined as follows:
X
nc h i
f cost ðfXgÞ ¼ ð1 þ ε1 :υÞε2 W ðfXgÞ, υ¼ max 0, gj ðfXgÞ ð16:2Þ
j¼1
where υ denotes the sum of the violations of the design constraints and nc is the
number of the constraints. Here, ε1 is set to unity and ε2 is calculated by
iter
ε2 ¼ 1:5 þ 1:5 ð16:3Þ
iter max
Thus, in the first steps of the search process, ε2 is set to 1.5 and ultimately
increased to 3. Such a scheme penalizes the infeasible solutions more severely as
the optimization process proceeds. As a result, in the early stages, the agents are
free to explore the search space, but at the end they tend to choose solutions with no
violation.
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300 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
This section describes the VPS algorithm. First, a brief overview of the free
vibration of single degree of freedom systems with viscous damping is provided,
and then the proposed method is presented.
There are two general types of vibrations, namely, free vibration and forced
vibration. In free vibration, the motion is only maintained by the restoring forces,
and in the forced vibration, a time-dependent force is applied to the system. The
effects of friction in a vibrating system can be neglected resulting in an undamped
vibration. However, all vibrations are actually damped to some degree by friction
forces. These forces can be caused by dry friction, or Coulomb friction, between
rigid bodies, by fluid friction when a rigid body moves in a fluid, or by internal
friction between the molecules of a seemingly elastic body. In this section, the free
vibration of single degree of freedom systems with viscous damping is studied. The
viscous damping is caused by fluid friction at low and moderate speeds. Viscous
damping is characterized by the fact that the friction force is directly proportional
and opposite to the velocity of the moving body [18].
Figure 16.1 shows the vibrating motion of a body or system of mass m having
viscous damping. A spring of constant k and a dashpot are connected to the block.
The effect of damping is provided by the dashpot, and the magnitude of the friction
force exerted on the plunger by the surrounding fluid is equal to cx_: (c is the
coefficient of viscous damping, and its value depends on the physical properties of
the fluid and the construction of the dashpot). When the block is displaced a
distance x from its position of stable equilibrium, the equation of motion can be
expressed as
Before presenting the solutions for this differential equation, we define the
critical damping coefficient cc as
cc ¼ 2mωn ð16:5Þ
rffiffiffiffi
k
ωn ¼ ð16:6Þ
m
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16.3 The Vibrating Particles System Algorithm 301
where ρ and φ are constants generally determined from the initial conditions of the
problem. ωD and ξ are damped natural frequency and damping ratio, respectively.
Equation (16.7) is shown in Fig. 16.2 and the effect of damping ratio on vibratory
motion is illustrated in Fig. 16.3.
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302 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
Fig. 16.3 Free vibration of systems with four levels of damping: (a) ξ ¼ 5 %, (b) ξ ¼ 10 %, (c)
ξ ¼ 15 %, and (d) ξ ¼ 20 %
methods, VPS has a number of individuals (or particles) consisting of the variables
of the problem. The solution candidates gradually approach to their equilibrium
positions that are achieved from current population and historically best position in
order to have a proper balance between diversification and intensification. In VPS,
the initial locations of particles are created randomly in an n-dimensional search
space.
where xij is the jth variable of the particle i, xmin and xmax are the minimum and the
maximum allowable variable bound vectors, and rand is a random number uni-
formly distributed in the range of [0, 1].
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16.3 The Vibrating Particles System Algorithm 303
For each particle, three equilibrium positions with different weights are defined,
and during each generation, the particle position is updated by learning from them:
(1) the historically best position of the entire population (HB), (2) a good particle
(GP), and (3) a bad particle (BP). In order to select the GP and BP for each
candidate solution, the current population is sorted according to their objective
function values in an increasing order, and then GP and BP are chosen randomly
from the first and second half, respectively.
A descending function based on the number of iterations is proposed in VPS to
model the effect of damping level in the vibration that is depicted in Fig. 16.3.
α
iter
D¼ ð16:11Þ
iter max
where iter is the current iteration number and itermax is the total number of iterations
for optimization process. α is a constant.
According to the above concepts, the update rules in the VPS are given by
xij ¼ w1 : D: A: rand1 þ HBj þ w2 : D: A: rand2 þ GPj
þ w3 : D: A: rand3 þ BPj ð16:12Þ
h i h i h i
A ¼ w1 : HBj xij þ w2 : GPj xij þ w3 : BPj xij ð16:13Þ
w1 þ w2 þ w3 ¼ 1 ð16:14Þ
where xij is the jth variable of the particle i; w1, w2, and w3 are three parameters to
measure the relative importance of HB, GP, and BP, respectively; and rand1,
rand2, and rand3 are random numbers uniformly distributed in the range of [0,1].
The effects of A and D parameters in Eq. (16.12) are similar to that of ρ and eξωn t
in Eq. (16.7). Also, the value of sin ðωD t þ φÞ is considered unity in Eq. (16.12)
(xðtÞ ¼ ρeξωn t are shown in Fig. 16.2 by red lines).
In order to have a fast convergence in the VPS, the effect of BP is sometimes
ignored in updating the position formula. Therefore, for each particle, a parameter
like p within (0, 1) is defined, and it is compared with rand (a random number
uniformly distributed in the range of [0,1]), and if p < rand, then w3 ¼ 0 and
w2 ¼ 1 w1.
There is a possibility of boundary violation when a particle moves to its new
position. In the proposed algorithm, for handling boundary constraints, a harmony
search-based approach is used [19]. In this technique, there is a possibility like
harmony memory considering rate (HMCR) that specifies whether the violating
component must be changed with the corresponding component of the historically
best position of a random particle or it should be determined randomly in the search
space. Moreover, if the component of a historically best position is selected, there is
a possibility like pitch adjusting rate (PAR) that specifies whether this value should
be changed with a neighboring value or not.
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304 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
In this chapter, after the predefined maximum evaluation number, the optimiza-
tion process is terminated. However, any terminating condition can be used.
Flowchart of the VPS is illustrated in Fig. 16.4.
This section discusses the computational examples used to investigate the perfor-
mance of the proposed algorithm. The values of the population size, the total
number of iteration, α, p, w1, and w2 are set to 20, 1500, 0.05, 70 %, 0.3, and 0.3
for all examples, respectively. Sensitivity analyses of the VPS on these parameters
are investigated in [17]. Twenty independent optimization runs are carried out for the
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16.4 Test Problems and Optimization Results 305
first four considered examples, and the last example has been solved five times
independently. The algorithm is coded in MATLAB, and the structures are ana-
lyzed using the direct stiffness method by our own codes.
The 10-bar plane truss is a well-known benchmark problem, and Fig. 16.5 shows
the topology and nodal and element numbering of this truss. The cross-sectional
area of each of the members is considered to be an independent variable. The
material density is 2767.99 kg/m3 and the modulus of elasticity is 68.95 GPa for all
elements. At each free node (1–4), a nonstructural mass of 453.6 kg is attached. The
range of cross-sectional area of all members is from 0.645 to 50 cm2. The first three
natural frequencies of the structure must satisfy the following limitations ( f1 7 Hz,
f2 15 Hz, and f3 20 Hz).
Table 16.1 provides a comparison between some optimal design reported in the
literature and the present work. It can be seen that the lightest design (i.e.,
530.77 kg) and the best standard deviation on average (i.e., 2.55 kg) are obtained
by the VPS. The firefly algorithm (FA) [12] achieved the best average optimized
weight (i.e., 535.07 kg), and after that the VPS obtained 535.64 kg. Table 16.2
reports the natural frequencies of the optimized structures, and it is clear that none
of the frequency constraints are violated. The VPS converges to the optimum
solution after 4620 analyses. The methods utilized by Lingyun et al. [9], and
Gomes [10] and Miguel and Fadel Miguel [12] give the best result in 8000, 2000,
and 50,000 analyses. However, the VPS achieve the best design of PSO [10] after
940 analyses.
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306 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
Table 16.1 Comparison of optimized designs found for the 10-bar plane truss problem
Areas (cm2)
Wang Lingyun Gomes Miguel and Fadel Present
Design variable et al. [7] et al. [9] [10] Miguel [12] work [1]
1 32.456 42.234 37.712 36.198 35.1471
2 16.577 18.555 9.959 14.030 14.6668
3 32.456 38.851 40.265 34.754 35.6889
4 16.577 11.222 16.788 14.900 15.0929
5 2.115 4.783 11.576 0.654 0.6450
6 4.467 4.451 3.955 4.672 4.6221
7 22.810 21.049 25.308 23.467 23.5552
8 22.810 20.949 21.613 25.508 24.4680
9 17.490 10.257 11.576 12.707 12.7198
10 17.490 14.342 11.186 12.351 12.6845
Weight (kg) 553.8 542.75 537.98 531.28 530.77
Average optimized N/A 552.447 540.89 535.07 535.64
weight (kg)
Standard deviation on N/A 4.864 6.84 3.64 2.55
average weight (kg)
Table 16.2 Natural frequencies (Hz) evaluated at the optimum designs of the 10-bar plane truss
problem
Natural frequencies (Hz)
Frequency Wang et al. Lingyun Gomes Miguel and Fadel Present
number [7] et al. [9] [10] Miguel [12] work [1]
1 7.011 7.008 7.000 7.0002 7.0000
2 17.302 18.148 17.786 16.1640 16.1599
3 20.001 20.000 20.000 20.0029 20.0000
4 20.100 20.508 20.063 20.0221 20.0001
5 30.869 27.797 27.776 28.5428 28.6008
6 32.666 31.281 30.939 28.9220 29.0628
7 48.282 48.304 47.297 48.3538 48.4904
8 52.306 53.306 52.286 50.8004 51.0476
The 37-bar plane truss with initial configuration is shown in Fig. 16.6. Nodal
coordinates in the upper chord and member areas are regarded as design variables.
In the optimization process, nodes of the upper chord can be shifted vertically. In
addition, nodal coordinates and member areas are linked to maintain the structural
symmetry. Thus, only five layout variables and fourteen sizing variables will be
considered for the optimization. All members on the lower chord (numbers 28–37)
are modeled as bar elements with constant rectangular cross-sectional areas of
4 103 m2, and the others are modeled as bar elements with initial cross-sectional
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16.4 Test Problems and Optimization Results 307
areas of 1 104 m2. The material density is 7800 kg/m3 and the modulus of
elasticity is 210 GPa for all elements. Nonstructural mass of 10 kg is attached to
each of the free nodes on the lower chord which remain fixed during the design
process. The first three natural frequencies of the structure must satisfy the follow-
ing limitations: f1 20 Hz, f2 40 Hz, and f3 60 Hz.
This truss structure was previously optimized by Wang et al. [7] utilizing an
evolutionary node shift method, Lingyun et al. [9] using niche hybrid genetic
algorithm, Gomes [10] employing particle swarm optimization algorithm, Miguel
and Fadel Miguel [12] using firefly algorithm, and Kaveh and Ilchi Ghazaan [15]
utilizing particle swarm optimization with an aging leader and challengers and
harmony search-based side constraint-handling approach. Table 16.3 presents a
comparison between the results of the optimal designs reported in the literature and
the present work. The best weight, average optimized weight, and standard devia-
tion on average weight obtained by VPS and HALC–PSO [15] are approximately
identical although their designs are different. Table 16.4 shows the optimized
structural frequencies (Hz) for various methods. None of the frequency constraints
are violated. The proposed method requires 7940 structural analyses to find the
optimum solution, while NHGA [9], PSO [10], FA [12], and HALC–PSO [15]
require 8000, 12,500, 50,000, and 10,000 structural analyses, respectively.
The 72-bar space truss is shown in Fig. 16.7 as the third design example. The
elements are divided into 16 groups, because of symmetry. The material density is
2767.99 kg/m3 and the elastic modulus is 68.95 GPa for all members. Four
nonstructural masses of 2268 kg are attached to the nodes 1 through 4. The
allowable minimum cross-sectional area of all elements is set to 0.645 cm2. This
example has two frequency constraints. The first frequency is required to be
f1 ¼ 4 Hz and the third frequency is required to be f3 6 Hz.
Optimal structures found by Konzelman [5], Gomes [10], Kaveh and Zolghadr
[11], Miguel and Fadel Miguel [12], and Kaveh and Ilchi Ghazaan [15] and the
proposed method are summarized in Table 16.5. The CSS–BBBC (hybridization of
charged system search and Big Bang with trap recognition capability) [11] obtained
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308 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
Table 16.3 Comparison of optimized designs found for the 37-bar truss problem
Y coordinates (m) and areas (cm2)
Miguel Kaveh
Wang and Fadel and Ilchi
et al. Lingyun Miguel Ghazaan Present
Design variable [7] et al. [9] Gomes [10] [12] [15] work [1]
Y3, Y19 (m) 1.2086 1.1998 0.9637 0.9392 0.9750 0.9042
Y5, Y17 (m) 1.5788 1.6553 1.3978 1.3270 1.3577 1.2850
Y7, Y15 (m) 1.6719 1.9652 1.5929 1.5063 1.5520 1.5017
Y9, Y13 (m) 1.7703 2.0737 1.8812 1.6086 1.6920 1.6509
Y11 (m) 1.8502 2.3050 2.0856 1.6679 1.7688 1.7277
A1, A27 (cm2) 3.2508 2.8932 2.6797 2.9838 2.9652 3.1306
A2, A26 (cm2) 1.2364 1.1201 1.1568 1.1098 1.0114 1.0023
A3, A24 (cm2) 1.0000 1.0000 2.3476 1.0091 1.0090 1.0001
A4, A25 (cm2) 2.5386 1.8655 1.7182 2.5955 2.4601 2.5883
A5, A23 (cm2) 1.3714 1.5962 1.2751 1.2610 1.2300 1.1119
A6, A21 (cm2) 1.3681 1.2642 1.4819 1.1975 1.2064 1.2599
A7, A22 (cm2) 2.4290 1.8254 4.6850 2.4264 2.4245 2.6743
A8, A20 (cm2) 1.6522 2.0009 1.1246 1.3588 1.4618 1.3961
A9, A18 (cm2) 1.8257 1.9526 2.1214 1.4771 1.4328 1.5036
A10, A19 (cm2) 2.3022 1.9705 3.8600 2.5648 2.5000 2.4441
A11, A17 (cm2) 1.3103 1.8294 2.9817 1.1295 1.2319 1.2977
A12, A15 (cm2) 1.4067 1.2358 1.2021 1.3199 1.3669 1.3619
A13, A16 (cm2) 2.1896 1.4049 1.2563 2.9217 2.2801 2.3500
A14 (cm2) 1.0000 1.0000 3.3276 1.0004 1.0011 1.0000
Weight (kg) 366.5 368.84 377.20 360.05 359.93 359.94
Average optimized N/A 378.8259 381.2 360.37 360.23 360.23
weight (kg)
Standard deviation on N/A 9.0325 4.26 0.26 0.24 0.22
average weight (kg)
Table 16.4 Natural frequencies (Hz) evaluated at the optimum designs of the 37-bar truss
problem
Natural frequencies (Hz)
Miguel and
Frequency Wang Lingyun Gomes Fadel Miguel Kaveh and Ilchi Present
number et al. [7] et al. [9] [10] [12] Ghazaan [15] work [1]
1 20.0850 20.0013 20.0001 20.0024 20.0216 20.0002
2 42.0743 40.0305 40.0003 40.0019 40.0098 40.0005
3 62.9383 60.0000 60.0000 60.0043 60.0017 60.0000
4 74.4539 73.0444 73.0440 77.2153 76.7857 77.2124
5 90.0576 89.8244 89.8240 96.9900 96.3543 97.3173
the lightest design; however, the best designs of all methods are approximately
identical. The average optimized weight and the standard deviation on average
weight of the VPS are less than those of all other methods. Frequency constraints
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16.4 Test Problems and Optimization Results 309
are satisfied by all methods (see Table 16.6). Figure 16.8 compares the best and
average runs of convergence histories for the proposed method. The VPS requires
4720 structural analyses to find the optimum solution, while PSO [10], FA [12], and
HALC–PSO [15] require 42,840, 100,000, and 8000 structural analyses,
respectively.
Figure 16.9 shows the 120-bar dome truss. The members are categorized into seven
groups because of symmetry. The material density is 7971.810 kg/m3, and the
modulus of elasticity is 210 GPa for all elements. Nonstructural masses are attached
to all free nodes as follows: 3000 kg at node one, 500 kg at nodes 2–13, and 100 kg
at the remaining nodes. Element cross-sectional areas can vary between 1 cm2 and
129.3 cm2. The frequency constraints are as follows: f1 9 Hz and f2 11 Hz.
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310
Table 16.5 Comparison of optimized designs obtained for the 72-bar truss problem
Areas (cm2)
16
Members in Konzelman Gomes Kaveh and Miguel and Fadel Kaveh and Ilchi Present
Design variable the group [5] [10] Zolghadr [11] Miguel [12] Ghazaan [15] work [1]
1 1–4 3.499 2.987 2.854 3.3411 3.3437 3.5017
2 5–12 7.932 7.849 8.301 7.7587 7.8688 7.9340
3 13–16 0.645 0.645 0.645 0.6450 0.6450 0.6450
4 17–18 0.645 0.645 0.645 0.6450 0.6450 0.6450
5 19–22 8.056 8.765 8.202 9.0202 8.1626 8.0215
6 23–30 8.011 8.153 7.043 8.2567 7.9502 7.9826
7 31–34 0.645 0.645 0.645 0.6450 0.6452 0.6450
8 35–36 0.645 0.645 0.645 0.6450 0.6450 0.6450
9 37–40 12.812 13.450 16.328 12.0450 12.2668 12.8175
10 41–48 8.061 8.073 8.299 8.0401 8.1845 8.1129
11 49–52 0.645 0.645 0.645 0.6450 0.6451 0.6450
12 53–54 0.645 0.645 0.645 0.6450 0.6451 0.6450
13 55–58 17.279 16.684 15.048 17.3800 17.9632 17.3362
14 59–66 8.088 8.159 8.268 8.0561 8.1292 8.1010
15 67–70 0.645 0.645 0.645 0.6450 0.6450 0.6450
16 71–72 0.645 0.645 0.645 0.6450 0.6450 0.6450
Weight (kg) 327.605 328.823 327.507 327.691 327.77 327.649
Average optimized weight N/A 332.24 N/A 329.89 327.99 327.670
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(kg)
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Standard deviation on aver- N/A 4.23 N/A 2.59 0.19 0.018
age weight (kg)
Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
16.4 Test Problems and Optimization Results 311
Table 16.6 Natural frequencies (Hz) evaluated at the optimum designs of the 72-bar truss
problem
Natural frequencies (Hz)
Kaveh and Miguel and Kaveh and Present
Frequency Konzelman Gomes Zolghadr Fadel Miguel Ilchi Ghazaan work
number [5] [10] [11] [12] [15] [1]
1 4.000 4.000 4.000 4.0000 4.000 4.0000
2 4.000 4.000 4.000 4.0000 4.000 4.0002
3 6.000 6.000 6.004 6.0000 6.000 6.0000
4 6.247 6.219 6.2491 6.2468 6.230 6.2428
5 9.074 8.976 8.9726 9.0380 9.041 9.0698
The comparison of the results of the VPS algorithm with the outcomes of other
algorithms is shown in Table 16.7. The present algorithm yields the least weight.
The best weight of the VPS algorithm is 8888.74 kg, while it is 9046.34 kg for CSS–
BBBC [11] and 8889.96 kg for the HALC–PSO [15]. Moreover, it can be seen that
the lightest average optimized weight and the standard deviation on average weight
are found by the proposed method. Table 16.8 reports the natural frequencies of the
optimized structures, and it is clear that none of the frequency constraints are
violated. Figure 16.10 compares the convergence curves of the best and the average
results obtained by the proposed method. The HALC–PSO [15] and VPS algo-
rithms get the optimal solution after 17,000 and 6860 analyses, respectively.
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312 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
The 600-bar single-layer dome structure shown in Fig. 16.11 is considered as the
last example. The entire structure is composed of 216 nodes and 600 elements. A
more detailed substructure is depicted in Fig. 16.12 to show the nodal numbering
and coordinates. Each of the elements of this substructure is considered as a design
variable. Thus, this is a size optimization problem with 25 variables. The material
density is 7850 kg/m3 and the elastic modulus is 200 GPa for all members.
A nonstructural mass of 100 kg is attached to all free nodes. The minimum
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16.4 Test Problems and Optimization Results 313
Table 16.7 Comparison of optimized designs obtained for the 120-bar dome problem
Areas (cm2)
Kaveh and Kaveh and Ilchi Present
Design variable Zolghadr [11] Ghazaan [15] work [1]
1 17.478 19.8905 19.6836
2 49.076 40.4045 40.9581
3 12.365 11.2057 11.3325
4 21.979 21.3768 21.5387
5 11.190 9.8669 9.8867
6 12.590 12.7200 12.7116
7 13.585 15.2236 14.9330
Weight (kg) 9046.34 8889.96 8888.74
Average optimized weight (kg) N/A 8900.39 8896.04
Standard deviation on average N/A 6.38 6.65
weight (kg)
Table 16.8 Natural frequencies (Hz) evaluated at the optimum designs of the 120-bar dome
problem
Natural frequencies (Hz)
Frequency Kaveh and Zolghadr Kaveh and Ilchi Ghazaan Present work
number [11] [15] [1]
1 9.000 9.000 9.0000
2 11.007 11.000 11.0000
3 11.018 11.000 11.0000
4 11.026 11.010 11.0096
5 11.048 11.050 11.0491
Fig. 16.10 Convergence curves obtained for the 120-bar dome truss
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314 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
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16.4 Test Problems and Optimization Results 315
Table 16.9 Comparison of optimized designs obtained for the 600-bar single-layer dome truss
problem
Areas (cm2)
Design variable (nodes) Kaveh and Ilchi Ghazaan [20] Present work [1]
1 (1–2) 1.4305 1.3030
2 (1–3) 1.3941 1.3998
3 (1–10) 5.5293 5.1072
4 (1–11) 1.0469 1.3882
5 (2–3) 16.9642 16.9217
6 (2–11) 35.1892 38.1432
7 (3–4) 12.2171 11.8319
8 (3–11) 16.7152 16.6149
9 (3–12) 12.5999 11.3403
10 (4–5) 9.5118 9.3865
11 (4–12) 8.9977 8.7692
12 (4–13) 9.4397 9.6682
13 (5–6) 6.8864 6.9826
14 (5–13) 4.2057 5.4445
15 (5–14) 7.2651 6.3247
16 (6–7) 6.1693 5.1349
17 (6–14) 3.9768 3.3991
18 (6–15) 8.3127 7.7911
19 (7–8) 4.1451 4.4147
20 (7–15) 2.4042 2.2755
21 (7–16) 4.3038 4.9974
22 (8–9) 3.2539 4.0145
23 (8–16) 1.8273 1.8388
24 (8–17) 4.8805 4.7965
25 (9–17) 1.5276 1.5551
Weight (kg) 6171.51 6133.02
Average optimized weight (kg) 6191.50 6142.03
Standard deviation on average weight (kg) 39.08 12.54
Table 16.10 Natural frequencies (Hz) evaluated at the optimum designs of the 600-bar single-
layer dome truss problem
Natural frequencies (Hz)
Frequency number Kaveh and Ilchi Ghazaan [20] Present work [1]
1 5.002 5.0000
2 5.003 5.0003
3 7.001 7.0000
4 7.001 7.0001
5 7.002 7.0002
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316 16 Vibrating Particles System Algorithm for Truss Optimization with Frequency. . .
Fig. 16.13 Convergence curves obtained for the 600-bar single-layer dome truss
frequencies of the optimized structures, and it is clear that none of the frequency
constraints are violated. The convergence rates of the best and average results found
by the proposed method are provided in Fig. 16.13. The ECBO and VPS algorithms
get the optimal solution after 19,020 and 19,740 analyses, respectively.
References
1. Kaveh A, Ilchi Ghazaan M (2016) Vibrating particles system algorithm for truss optimization
with multiple natural frequency constraints. Acta Mech. doi:10.1007 /s00707-016-1725-z.
First Online: 20 Sept 2016
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References 317
2. Tong WH, Jiang JS, Liu GR (2000) Solution existence of the optimization problem of truss
structures with frequency constraints. Int J Solids Struct 37:4043–4060
3. Bellagamba L, Yang TY (1981) Minimum mass truss structures with constraints on funda-
mental natural frequency. AIAA J 19:1452–1458
4. Lin JH, Chen WY, Yu YS (1982) Structural optimization on geometrical configuration and
element sizing with static and dynamic constraints. Comput Struct 15:507–515
5. Konzelman CJ (1986) Dual methods and approximation concepts for structural optimization.
MSc Thesis, Department of Mechanical Engineering, University of Toronto, Canada
6. Grandhi RV, Venkayya VB (1988) Structural optimization with frequency constraints. AIAA J
26:858–866
7. Wang D, Zha W, Jiang J (2004) Truss optimization on shape and sizing with frequency
constraints. AIAA J 42:622–630
8. Sedaghati R (2005) Benchmark case studies in structural design optimization using the force
method. Int J Solids Struct 42:5848–5871
9. Lingyun W, Mei Z, Guangming W, Guang M (2005) Truss optimization on shape and sizing
with frequency constraints based on genetic algorithm. Comput Mech 35:361–368
10. Gomes HM (2011) Truss optimization with dynamic constraints using a particle swarm
algorithm. Expert Syst Appl 38:957–968
11. Kaveh A, Zolghadr A (2012) Truss optimization with natural frequency constraints using a
hybridized CSS–BBBC algorithm with trap recognition capability. Comput Struct
102–103:14–27
12. Miguel LFF, Fadel Miguel LF (2012) Shape and size optimization of truss structures consid-
ering dynamic constraints through modern meta-heuristic algorithms. Expert Syst Appl
39:9458–9467
13. Zuo W, Bai J, Li B (2014) A hybrid OC–GA approach for fast and global truss optimization
with frequency constraints. Appl Soft Comput 14:528–535
14. Kaveh A, Javadi SM (2014) Shape and size optimization of trusses with multiple frequency
constraints using harmony search and ray optimizer for enhancing the particle swarm optimi-
zation algorithm. Acta Mech 225:1595–1605
15. Kaveh A, Ilchi Ghazaan M (2015) Hybridized optimization algorithms for design of trusses
with multiple natural frequency constraints. Adv Eng Softw 79:137–147
16. Hosseinzadeh Y, Taghizadieh N, Jalili S (2016) Hybridizing electromagnetism-like mecha-
nism algorithm with migration strategy for layout and size optimization of truss structures with
frequency constraints. Neural Comput Appl 27(4):953–971
17. Kaveh A, Ilchi Ghazaan M (2016) A new meta-heuristic algorithm: vibrating particles system.
Scientia Iranica (Published online)
18. Beer FP, Johnston ER Jr, Mazurek DF, Cornwell PJ, Self BP (2013) Vector mechanics for
engineers. McGraw-Hill Companies, New York, NY
19. Kaveh A, Talatahari S (2010) A novel heuristic optimization method: charged system search.
Acta Mech 213:267–286
20. Kaveh A, Ilchi Ghazaan M (2016) Optimal design of dome truss structures with dynamic
frequency constraints. Struct Multidiscip Optim 53(3):605–621
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Chapter 17
Cost and CO2 Emission Optimization
of Reinforced Concrete Frames Using
Enhanced Colliding Bodies Optimization
Algorithm
17.1 Introduction
mass, and long service life, contributes 5 % of annual anthropogenic global CO2
production. Main contributor for it to happen is chemical conversion process used
in the production of Portland clinker and cement production by fossil fuel combus-
tion. With annual consumption approaching 20,000 million metric tons of concrete,
the manufacturing process releases 0.9 tons of CO2 per ton of clinker [2]. In
addition to the 1.6 billion tons of cement used worldwide, the concrete industry is
consuming 12.6 billion tons of raw materials each year. Thus, besides cement’s role
in CO2 emission, mining, processing, and transporting of raw materials consume
energy in large quantities and adversely affect theology of the planet [4]. Reducing
atmospheric concentration of CO2 caused by construction industry can be reached
through innovative architecture, sustainable structural design, and reducing the
cement of concrete mixture [2].
The purpose of this chapter is to present an optimal design technique in order to
achieve more sustainable, environmentally friendly, and economically feasible
structural design. The methods of structural optimization can be divided into two
categories: exact methods and approximate methods. The exact methods are based
on mathematical programming techniques such as the Lagrangian multipliers
method, convex programming, linear programming, and sequential unconstrained
minimization for which the required computational cost for finding an optimal
solution grow polynomially with problem size, hence the applications of the exact
methods are limited to simple and deterministic polynomial problem instances. To
overcome these problems, metaheuristic methods are developed. These methods
provide the practical possibility to improve the design process without the need for
complex analysis; however, they require a great computational effort because of a
large number of iterations needed for the evaluation of objective functions and
structural constraints.
Some recent research studies are focused on cost optimization of reinforced
concrete structures using evolutionary optimization methods. Rajeev and
Krishnamoorthy [5] applied a simple genetic algorithm to perform optimal design
of planar reinforced concrete frames, Camp et al. [6] used genetic algorithm for
flexural design of RC frames, Lee and Ahn [7] applied genetic algorithm to
optimum design of two-dimensional frames, Paya-Zaforteza et al. [8] conducted a
multi-objective comparison for RC building frames using simulated annealing,
Kwak and Kim [9] studied an optimum design of RC plane frames using integrated
genetic algorithm complemented with direct search, Kaveh and Sabzi [10]
conducted a comparative study of heuristic big bang–big crunch, heuristic particle
swarm, and ant colony optimization for optimum design of RC frames, and Akin
and Saka [11] used harmony search algorithm for optimum detailed design of RC
plane frames.
Recently, attention to the preservation of environment and reducing CO2 emis-
sions has been the focus of studies in optimum design of RC structures. Paya-
Zaforteza et al. [12] used simulated annealing for CO2 optimization of reinforced
concrete frames; Camp and Huq [13] applied the Big Bang–Big Crunch algorithm
for CO2 and cost optimization of RC frames. The objective of this chapter is
optimal design of cost and CO2 emissions in terms of cross-section dimensions
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17.2 Formulation of the RC Frame Optimization Problem 321
The assessment of the objective functions requires the definition of the structure in
terms of the design variables including cross-sectional dimensions of elements, area
and type of steel bars, and resisting capacity. Due to the discreteness of member
dimensions and reinforcement sizes, large number of sections, and different pat-
terns of reinforcements, two section databases for beams and columns are created to
reduce the elaboration of the problem. The identification numbers of the sections
are related with all design variables. It is worth pointing out that the capacity of
members is defined by applying ultimate strength design method. Two section
databases are created based on ACI building code criteria and specified assump-
tions, which are followed for both beams and columns sections.
17.2.1.1 Beams
For beams, the sections are considered as rectangular and singly reinforced; there-
fore, the compression reinforcement at support and the tension reinforcement near
mid-span are checked separately. This approach leads to a conservative and simple
analysis. The area of steel varies from one #3 bar to a maximum of four #11 bars.
The depth to width ratio varies between 1 and 2.5.
The last distance measured from the surface of the concrete member to the
surface of the embedded reinforcing steel is taken as 380 mm. The assumed ranges
and increment steps for cross-sectional dimensions are different in each design
example. Figure 17.1a defines the geometry of a general rectangular singly
reinforced concrete beam.
To evaluate flexural response of the beam elements, their capacity is defined
using the ACI code. In order to ensure ductile failure, these must be designed as
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322 17 Cost and CO2 Emission Optimization of Reinforced. . .
di
d d
h h
tc tc
td td
b b
As f y
a¼ 0 ð17:2Þ
0:85f c b
where f 0c denotes the specified compressive strength of concrete and b is the width of
section.
Taking the abovementioned rules into account, DB sections for beams
containing the width, the height, the number of reinforcing bar, the steel ratio, the
moments of inertia, and the ultimate bending moment capacity can be created.
Finally, the sections are arranged in the order of increasing moment resisting
capacities.
17.2.1.2 Columns
For columns, the sections are considered as rectangular tied and short, so the
applied moment will not be magnified. The area of steel varies from four #3 bars
to a maximum of twelve #11 bars. For the rebar topologies, an even number of bars
with the same size are distributed along all four faces so that the column is
symmetric about the axis of bending. Table 17.1 represents the prespecified
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17.2 Formulation of the RC Frame Optimization Problem 323
reinforcement patterns for columns. The depth to width ratio is considered between
1 and 2.5. Figure 17.1b defines the geometry of a rectangular tied column.
Column sections are subjected to bending moment in combination with axial
forces; therefore, the equilibrium of internal forces changes resulting in different
behavioral modes depending on the level of accompanying eccentricity. The
sustainability and serviceability of column sections can be evaluated in a variety
of combinations of bending moment and axial force derived by varying the applied
axial strain. To find points corresponding to a specific value of strain distribution
within the cross section, a rectangular stress block in the concrete must be deter-
mined. The same method is used to specify the stress distribution in reinforcement.
Plotting values of load and moment capacities corresponding to different assumed
values for the neutral axis depth (resulting in different strain distributions) via an
iterative calculation results in some contour charts called interaction diagrams.
Figure 17.2 shows a curve plot of controlling key points connected by linear
relationships for a typical column section. The nominal axial load capacity for a
given strain distribution defined by ACI Code is found by
Xn
Pn ¼ Cc þ i¼1
Fsi ð17:3Þ
Cc ¼ 0:85f 0c ab ð17:4Þ
where Es is the elastic modulus of reinforcement and εsi is the strain of the ith layer
of steel given as
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324 17 Cost and CO2 Emission Optimization of Reinforced. . .
εs = εy
εs > εy
M0 Mn
c di
εsi ¼ 0:003 ð17:7Þ
c
where c is:
0:003
c¼ di ð17:8Þ
0:003 εy
The nominal moment capacity for the specified strain distribution defined by
ACI Code is found by
X
h a n h
M n ¼ Cc þ F
i¼1 si 2
di ð17:9Þ
2 2
where a is:
a ¼ β1 c ð17:10Þ
and β is:
f 0c 30
β1 ¼ 0:85 0:05 0:65 if f 0c > 30 MPa ð17:11aÞ
7
β1 ¼ 0:85 if 30 MPa < f 0c < 50 MPa ð17:11bÞ
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17.2 Formulation of the RC Frame Optimization Problem 325
where x is the vector of design variables that are taken as the area of steel and the
geometry of cross sections of beams and columns, Ci is the normalized degree of
violation of the ith constraint, n is the number of constraints, and k > 0 is a penalty
exponent required for tuning the penalty function. Since k reflects the solution
quality, imposing a large k results in severe penalty, which is reflected in rapid
convergence to local optima (exploitation). Conversely, a small k reduces the
severity of penalty; therefore, a comprehensive search through the search space
with slow convergence will be used to explore the solutions (exploration). Depending
on the case study, penalty exponent can be obtained through trial and error.
Structural capacity of reinforced concrete beams must be greater than the ultimate
bending moment derived from the applied loading. The moment capacity penalty
can be expressed in normalized form as below:
jMu j ∅Mn
C1 ¼ ð17:13Þ
∅Mn
where Mu is the ultimate applied moment and ∅ is the strength reduction factor. For
compression-controlled sections having a net tensile strain in the extreme tension
steel equal to or smaller than 0.002 while the extreme fibers of compression face in
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326 17 Cost and CO2 Emission Optimization of Reinforced. . .
concrete reach its crushing strain of 0.003, ∅ is taken as 0.65, and for tension-
controlled sections having the strain values in tension reinforcement farthest from
the compression face of a member >0.005 while concrete reaches its crushing
strain of 0.003, ∅ is taken as 0.9. Sections between these two extremes are called
transition sections, and the strength reduction factor is calculated by linear
interpolation.
In order to prevent the possibility of sudden failure and improve the cracking
behavior, the lower bound of reinforcement ratio is limited to
pffiffiffiffi0
f c 1:4
ρmin ¼ ð17:14Þ
4f y fy
C2 ¼ ρmin ρ ð17:15Þ
To ensure the ductile behavior and the requirements for placing the reinforcing
bars, the upper bound on the reinforcement ratio is limited to
f 0c 600
ρmax ¼ 0:85β1 ð17:16Þ
f y 600 þ f y
C3 ¼ ρ ρmax ð17:17Þ
l
hmin ¼ ð17:18Þ
21
where l is the span of the beam. The penalty for the thickness of the beam can be
expressed as
hmin h
C4 ¼ ð17:19Þ
hmin
If the rectangular compression-block depth is greater than the effective depth, the
penalty is applied as
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17.2 Formulation of the RC Frame Optimization Problem 327
ad
C5 ¼ ð17:20Þ
d
In order to place and compact concrete between bars satisfactorily and provide
proportionate bond, the minimum clear spacing smin should be db but not <1 in.
Here db is the diameter of reinforcement bars. The bar spacing penalty is
smin s
C6 ¼ ð17:21Þ
smin
Since the section capacities are evaluated separately, the reinforcement topology
including bar spacing and steel ratio could be different in both sections at the
support and mid-span while the dimensions are the same. For this reason, the same
procedure for determining constraints related to reinforcement topology must be
performed for the section under negative bending moment.
A column section is acceptable when the design action effects defined by combi-
nation of Mn and Pn fall within the load–moment interaction diagram. The load–
moment interaction penalty can be expressed as
r r0
C7 ¼ ð17:22Þ
r0
where r is the radial distance between the origin of the interaction diagram and the
corresponding pair under the applied loading and r 0 is the radial distance between
the origin of the interaction diagram and the intersection of vector r with the load–
moment curve.
For compression members, the minimum longitudinal reinforcement ρmin is
limited to 0.01. The minimum reinforcement penalty is
C8 ¼ ρmin ρ ð17:23Þ
C9 ¼ ρ ρmax ð17:24Þ
The clear distance between longitudinal bars should be 1.5 db but not <1.5 in. The
longitudinal bar spacing penalty is
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328 17 Cost and CO2 Emission Optimization of Reinforced. . .
smin s
C10 ¼ ð17:25Þ
smin
Since the bars are distributed along all four faces, the longitudinal bar spacing
constraint must be checked in both width side and height side of the section.
The optimal design criterion for reinforced concrete frames involves two different
objective functions: The first objective function is based on the most economical
solution that accounts for the cost of materials in terms of the concrete, the steel,
and the labor cost in construction process. The second objective function quantifies
the embedded CO2 resulting from the use of materials, which involve emissions at
different stages of the production and the placement of concrete and steel in
structure. The unit costs and CO2 emissions were obtained from the 2007 database
of the Institute of Construction Technology of Catalonia [17]. It is important to note
that the calculation of GHG or CO2 emissions of buildings does not contain
transport emissions including transportation for building materials, construction
equipment, and workers, since transport distance from cradle to site is highly
dependent on the case study. The general form of the objective function for current
study can be expressed as
X
n
min : f ðxÞ ¼ ui m i ð x 1 ; x 2 ; . . . ; x r Þ
i¼1
ð17:26Þ
s:t: Ci ðx1 ; x2 ; . . . ; xr Þ 0
where ui represents the unit prices or unit CO2 emissions of material and
construction components, mi is the measurements of the construction units, xi are
the design variables, n is the number of construction members, r is the number of
design variables, and Ci (i ¼ 1, 2, . . ., n) are the design constraints.
Metaheuristic algorithms are often based on the simulation of natural evolution and
the principle of preservation or the survival of the fittest, which is a hypothetical
population-based optimization procedure. In other words, a metaheuristic algorithm
is an iterative process, which applies a set of agents to move through the design
space and seek near-optimal solutions of the complex problems in a reasonably
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17.3 Formulation of the Optimization Problem 329
Collision is a short-term interaction between two bodies in which they are pushed
away from each other and tend to form the most stable configuration and achieve
the lowest energy state. According to the law of energy and momentum conserva-
tion, in all collisions the total amount of momentum possessed by the two objects
does not change, i.e., the amount of momentum gained by one object is equal to the
amount of momentum lost by the other object while the total kinetic energy after the
collision may not be equal to the total kinetic energy before the collision and it
changes to some other form of energy. What distinguishes different types of
collisions is whether they conserve kinetic energy. When the total kinetic energy
of system is lost, a perfectly inelastic collision occurs in which the two bodies stick
together after the impact. Contrariwise if the total kinetic energy of system is
conserved, a perfectly elastic collision occurs. The plot for this configuration is
shown in Fig. 17.3.
In terms of this conception, the search ability of the CBO algorithm can be
framed based on the interaction between colliding bodies (CBs) that are moving
through predefined amplitude, starting with random initial positions to find near-
optimal solutions. Each colliding body, as a solution candidate, contains a number
of decision variables and is characterized by its position and velocity. The laws of
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330 17 Cost and CO2 Emission Optimization of Reinforced. . .
where fit is the objective function value of the CBs and n is an even number of
colliding bodies. In order to select pairs of objects for collision, CBs are sorted
according to the value of their objective function in an increasing order and divided
into two equal groups. Agents with upper fitness values (moving objects) and finite
speed push the corresponding agents with lower fitness values (stationary objects),
which are at rest before the collision, toward better positions. The velocity of
moving bodies before the collision is given as
n
vi ¼ xi xin2 i¼ þ 1, . . . , n ð17:28Þ
2
where xi is the position vector of the ith CB in moving group and xin2 is the
corresponding position vector in the stationary group.
After the collision, the attributes of each moving object are updated as follows:
mi εmin2 vi n
v0i ¼ i ¼ þ 1 , ... , n ð17:29Þ
mi þ min2 2
x0i ¼ xin2 þ rv0i ð17:30Þ
where mi is the mass of the ith moving CB, vi is the velocity of the ith moving CB
before the collision, min2 is the mass of the ith stationary CB, xin2 is the previous
position of the ith stationary CB, r is a random vector uniformly distributed in the
range of (1,1), and ε represents the coefficient of restitution defined as
iter
ε¼1 ð17:31Þ
iter max
where iter is the number of iterations. Adjustment of this indicator changes the rate
of intensification and diversification in the system and generally ranges between
zero and one.
In addition, the attributes of each stationary object after the collision, which now
has a velocity in the same direction of the moving object, are updated as follows:
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17.3 Formulation of the Optimization Problem 331
miþn2 þ εmiþn2 viþn2 n
v0i ¼ , i ¼ 1 , ... , ð17:32Þ
mi þ miþn2 2
x0i ¼ xi þ rv0i ð17:33Þ
where mi is the mass of the ith stationary CB, miþn2 is the mass of the ith moving CB,
viþn2 is the velocity of the ith moving CB before the collision, and xi is the previous
position of the ith stationary CB.
Historical best solutions are saved by employing the colliding memory
(CM) which stores some best solutions of each iteration found in previous popula-
tion and substitutes them with some current worst CB vectors. Introducing new best
bodies into the population prevents the population from moving only to neighbor-
ing states and speeds up the convergence rate without increasing the computational
cost.
In order to break one or more members of the population out of local minima and
produce a more efficient search, one component of the ith CB is regenerated in a
random manner in any given generation. The probability of choosing the compo-
nent is expressed as Pro, which ranges between (0, 1).
In accord with the given definition, enhanced colliding bodies algorithm is a
continuous variable-based method improved by saving the best solutions and
regenerating random members of population occasionally to produce a more
efficient and reliable solution. The steps of this algorithm can briefly be outlined
as follows:
Step 1: Randomly initialize the vector of CBs with n variables and evaluate their
associated fitness function.
Step 2: Store some best solutions of each iteration in the colliding memory and
replace them with the current worst CB vectors.
Step 3: Calculate the mass value for each CB using Eq. (17.27).
Step 4: Sort the fitness value of the objective function for each CB in an increasing
order, and then determine the pairs of CBs for collision.
Step 5: Evaluate the velocity of moving bodies before the collision using
Eq. (17.28).
Step 6: Update the velocities of stationary and moving bodies after the collision
using Eqs. (17.32) and (17.29), respectively.
Step 7: Update the positions of stationary and moving bodies using the generated
velocities after the collision in Step 6 and Eqs. (17.33) and (17.30), respectively.
If some bodies’ new positions violate the boundaries, correct their position and
return to the specified domain.
Step 8: Compare Pro with a random number, rni (i ¼ 1, 2 . . . n), which is distributed
uniformly between (0, 1), if rni < pro, randomly select a CB from both moving
and stationary group and regenerate one related component accidentally.
Step 9: Return to Step 2 until a terminating criterion is satisfied.
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332 17 Cost and CO2 Emission Optimization of Reinforced. . .
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17.4 Design Examples 333
Figure 17.4 illustrates the two-bay six-story frame originally designed by Rajeev
and Krisnamoorthy [5] using standard GA algorithm and redesigned by Camp at
al. [6, 13] using GA and BB–BC algorithm. The height of each story is 4 m and the
span of the left and right bay is 6 m and 4 m, respectively. The optimal dimension of
width for beam and column sections is considered between (200, 460) mm and
(150, 560) mm, respectively. The step of increment for both beam and column
sections is 30 mm. As shown in Fig. 17.4, the frame consists of 12 beams and
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334 17 Cost and CO2 Emission Optimization of Reinforced. . .
where Cc is the unit cost of concrete, Cs is the unit cost of steel reinforcement, Asi is
the area of reinforcing bars, Cf is the unit cost of formwork, nb is the number of
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17.4 Design Examples 335
beams, and nc is the number of columns. The unit costs of concrete, steel, and
formwork are estimated as $735=m3 , $7:1 =kg, and $54=m2 , respectively.
Table 17.3 compares the results obtained by the proposed algorithm with the
previous solutions.
The best solution reported by the ECBO is 23,081.57$. The best ECBO design is
2.46 % less than the best solution given by BB–BC.
Five more types of grouping are considered for the design of frame listed in
Table 17.2. The comparison of the solutions (Fig. 17.5) shows a maximum of
4.39 % decrease in cost for case 2 of grouping. Since the members in the same
group have the same design variables, the capacity violations must be relatively
close. More precisely, the internal force distributions in each group, which is highly
related to the load pattern, should have insignificant difference as much as possible.
Hence, the pattern of grouping should match closer to the internal force distribu-
tions while the number of groups should compromise between the economic design
and computing time. The information pertaining to compare the strength ratio
between different cases of grouping has been quantified in Fig. 17.6.
One of the best approaches to handle the constraints is evaluating the fitness
function in the feasible search space. This approach is called death penalty. The
feasible region is achieved by rejection of infeasible individuals. Some of the
geometric constraints can be applied during the process of creating DB sections.
Therefore, no further calculations are necessary to enforce these constraints on the
objective function. This technique is limited to problems in which the constraints
are not dependent on the geometric information related to the structure. The
remaining constraints to be checked in each iteration are the capacity ( C1 , C7
and the allowable thickness (C4 restrictions. Taking the abovementioned procedure
into account, the size of the search space is declined to 7.28e24 (Table 17.4). The
algorithm could attain the similar best solution in a significant short iteration
number of 800 and computational time of 0.46 s which is 6.93 times faster than
case 1. With the stopping criterion of 3000, it could decrease the solution by 2.73 %
with the computational time of 2.56 s, which is 1.24 times faster than case 1. As
shown in Fig. 17.7, the speed of convergence to the optimum value has had a
considerable increase.
Figure 17.8 illustrates the two-bay four-story frame originally designed by Paya-
Zaforteza et al. [8] using SA algorithm and redesigned by Camp et al. [13] using
BB–BC algorithm. The height of each story is 3 m, and the span of each bay is 5 m.
The optimal dimension of width for beam and column sections is considered
between (150, 1200) mm and (250, 1200) mm, respectively. The step of increment
for beam sections is 10 mm and for column sections is 50 mm. As shown in
Fig. 17.8, the frame is consisted of 8 beams and 12 columns arranged in 4 beam
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336
Beam 1 280 560 2#6+2#8 360 480 3#5+1#10 230 530 2#6+1#8
2 330 480 1#5+2#7 330 430 1#9+1#10 200 370 1#6+1#8
3 230 560 4#4+1#11 200 480 2#6+2#9 200 490 1#8+1#11
4 200 480 1#6+2#5 230 330 2#5+2#6 200 430 3#4+2#7
Column 1 180 200 4#5 180 280 4#5 180 270 4#4
2 180 460 4#7 280 250 8#5 210 330 4#5
3 180 280 4#4 150 200 6#3 210 360 6#4
Best Cost ($) 24,959 23,664 23,081.57
Average ($) – 26,520.55 27,028.98
Std deviation ($) – 1069.91 2695.02
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Cost and CO2 Emission Optimization of Reinforced. . .
17.4 Design Examples 337
24,400
24,200
24,000
Cost (dollar)
23,800
23,600
23,400
23,200
23,000
22,800
22,600
1 2 3 4 5 6
Grouping
5
Grouping
2
Beam
1 Column
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Strength Ratio
Fig. 17.6 Strength ratio in the groups for different cases of grouping
groups and 8 column groups. The spacing considered between adjacent parallel
frames is 5.00 m, and the thickness of the slab for all stories is 290 mm. Twelve load
combinations that include counteracting effects of dead, live, and wind loads are
taken into account to determine the required strength of the members as listed
below:
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338 17 Cost and CO2 Emission Optimization of Reinforced. . .
110,000 Case 1
100,000 Case 2
Case 3
90,000
Cost (dollar)
80,000
70,000
60,000
50,000
40,000
30,000
20,000
0 500 1000 1500 2000 2500 3000
Iteration
Fig. 17.7 Convergence rate in different size of the search space and number of iteration [1]
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17.4 Design Examples 339
U ¼ 1:5D ð17:35aÞ
U ¼ 1:5D þ 1:6L1 ð17:35bÞ
U ¼ 1:5D þ 1:6L2 ð17:35cÞ
U ¼ 1:5D þ 1:6LT ð17:35dÞ
U ¼ 1:5D þ 1:6W1 ð17:35eÞ
U ¼ 1:5D þ 1:6W2 ð17:35fÞ
U ¼ 1:5D þ 1:44L1 þ 1:44W1 ð17:35gÞ
U ¼ 1:5D þ 1:44L2 þ 1:44W1 ð17:35hÞ
U ¼ 1:5D þ 1:44LT þ 1:44W1 ð17:35iÞ
U ¼ 1:5D þ 1:44L1 þ 1:44W2 ð17:35jÞ
U ¼ 1:5D þ 1:44L2 þ 1:44W2 ð17:35kÞ
U ¼ 1:5D þ 1:44LT þ 1:44W2 ð17:35lÞ
where D is the uniform dead load applied to each beam, L1 stands for the live load
applied to only one beam in each story while the bays change alternatively, L2 is the
uniform live load applied in a pattern opposite of L1, W1 is the wind load applied to
the left side of the frame, and W2 is the wind load applied to the right side of the
frame. Table 17.5 lists the values of the uniform loads and wind loads at each story.
Compressive strength of concrete varies in each story from 25 to 50 MPa with the
increment step of 5 MPa. The unit weight of concrete is 2323 kg=m3 . Reinforce-
ment has the yield strength of 500 MPa, and the unit weight of 7849 kg=m3 . The
number of DB sections created for beams and columns are 98,424 and 7584,
respectively, which results in a design space of 2.23e60. The frame has a total of
60 design variables. Hence, the population of 16 CBs with a typical stopping
criterion of 4000 was required. In this example, two objective functions are
implemented to minimize cost and CO2 emissions in terms of the materials and
construction process. The general form of the cost function is defined as
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340 17 Cost and CO2 Emission Optimization of Reinforced. . .
Xnbþ nc Xnb
fk ¼ i¼1
fCc bi hi þ Cs Asi gli þ i¼1
fCf ðbi þ 2ðhi ti ÞÞ þ Ct bi gli
Xnc
þ i¼1
f2Cf ðbi þ hi Þgli ð17:36Þ
where Ct is the unit rate of scaffolding and ti is the thickness of the slab. The CO2
emission function has the same form of the cost function; however, the unit values
are different and also the scaffolding term is not considered. The unit rates for cost
and CO2 emissions are listed in Table 17.6.
The results for single objective of cost function obtained by the proposed
algorithm and the previous research works are compared in Table 17.7. The best
solution reported by the ECBO is 3429.92€ with 3587.88 kg of CO2 emissions. The
best ECBO cost design is 3.13 % less than the best solution given by BB–BC.
Concrete represents 18.22 % of the total cost, while reinforcing steel about 25.55 %
of the total cost. Table 17.8 compares the results for single objective of CO2
emission functions. The best solution reported by the ECBO is 3238.25 kg with a
cost of 3525.27€. The best ECBO CO2 design is 2.67 % less than the best solution
given by BB–BC. The percentage comparison of the solutions indicates that the
best CO2 emission design decreased the CO2 emissions by 9.74 % with a slight
increase in cost of 2.77 %. Since more environmentally friendly solutions are
recommended by IPCC, on the other hand, the low-CO2 emission design could
decrease the CO2 emissions considerably at an acceptable cost increment in prac-
tice; it seems that designing the RC structures based on the CO2 emissions is more
logistical (Table 17.9).
Figure 17.9 compares the strength ratio in element groups for both cost and CO2
objective functions. As can be seen, in beam groups the use of section capacity in
low-cost design is lower than low-CO2 emission design, while in column groups the
use of section capacity is higher. This finding shows that there is a relationship
between the geometry of frame and the objective functions. Table 17.2 indicates the
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17.4
Table 17.7 Design results for cost objective for two-bay four-story frame
BB–BC [13] Kaveh and Ardalani [1]
Concrete Concrete
Design Examples
Member type Group no. strength (MPa) Width (mm) Depth (mm) Bars strength (MPa) Width (mm) Depth (mm) Bars
Beam 1 40 180 430 1#8+2#8 30 220 430 2#7+3#7
2 40 180 450 1#10+2#8 30 250 450 2#7+4#5
3 30 190 460 1#8+1#11 30 220 440 3#6+3#6
4 25 220 530 4#4+1#10 25 220 430 1#9+3#7
Column 1 40 250 550 6#5 30 300 500 8#3
2 40 250 300 4#5 30 300 400 6#4
3 30 250 300 4#6 30 250 350 8#3
4 25 250 300 6#6 25 250 350 12#4
5 40 250 300 4#5 30 300 450 6a#4
6 40 250 250 8#5 30 250 250 4#4
7 30 250 250 6#4 30 250 300 4#3
8 25 250 250 4#3 25 250 250 4#3
Best cost (€) 3540.88 3429.92
Average (€) 3790.25 3682.09
Std deviation (€) 139.28 156.51
CO2 emission (kg) 3778.24 3587.88
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341
342
Table 17.8 Design results for CO2 objective for two-bay four-story frame
BB–BC [13] Kaveh and Ardalani [1]
Concrete Concrete
Member type Group no. strength (MPa) Width (mm) Depth (mm) Bars strength (MPa) Width (mm) Depth (mm) Bars
Beam 1 50 210 510 2#5+3#6 40 230 420 1#8+3#7
2 30 220 530 2#5+1#10 40 240 510 4#4+3#6
3 25 210 520 2#5+2#7 25 250 550 4#4+3#5
17
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Cost and CO2 Emission Optimization of Reinforced. . .
17.4 Design Examples 343
1.06
0.91
0.6
0.5 1.01
0.4
0.3
0.2
0.1
0
1 2 3 4 5 6 7 8 9 10 11 12
Element Groups
Fig. 17.9 Strength ratios in element groups for both cost and CO2 objective functions
ratio between cost and CO2 -optimized design variables. The dimension of beams
are bigger over the low-CO2 emission design than over the low-cost design.
In Table 17.10, the percentage of cost and CO2 emissions is quantified for
materials and construction components. Concrete, reinforcing steel, formwork, and
scaffolding represent approximately 26, 18, 46, and 10 % of the total cost and
50, 35, and 15 % of the total emissions, respectively.
Table 17.11 summarizes the results of the ECBO single-objective and multi-
objective designs. The best NSECBO design with lower cost is 3490€ with 3475 kg
of CO2 emissions which are 1.78 % and 7.31 % higher compared to single-objective
designs of cost and CO2 emissions, respectively. Alternatively, the best NSECBO
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344 17 Cost and CO2 Emission Optimization of Reinforced. . .
3440
3420
Co2 (kg)
3400
3380
3360
3340
3320
3300
3490 3495 3500 3505 3510 3515 3520
Cost (euro)
design with lower emissions is 3318 kg with a cost of 3520€, which are 2.47 % and
2.65 % higher, respectively. Both objectives are closely related and result in similar
solutions. All these lead to a tentative conclusion that the CO2 and cost objectives
should be considered together in RC structural designs. The Pareto front is
presented in Fig. 17.10.
Figure 17.11 illustrates the two-bay six-story frame originally designed by Paya-
Zaforteza et al. [8] using SA algorithm and redesigned by Camp et al. [13] using
BB–BC algorithm. The story height and bay span of the frame and the search space
specifications are the same as defined for the two-bay four-story frame in Example
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17.4 Design Examples 345
2. As shown in Fig. 17.11, the frame consists of 12 beams and 18 columns, which
are arranged in 6 beam groups and 12 column groups. The type of grouping, spacing
considered between adjacent parallel frames, the thickness of the slab, the strength
and the unit weight of concrete and steel, the load patterns, and the magnitude of
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346 17 Cost and CO2 Emission Optimization of Reinforced. . .
loads except the wind loads are the same as in Example 2. Table 17.12 lists the
values of the wind loads at each story. The frame has a total of 90 design variables
and the design space of 3.34e90. The general form of the objective functions is
given in Eq. (17.36).
Table 17.13 compares the results for single objective of cost function obtained
by the proposed algorithm with those of the previous researches. The best solution
reported by the ECBO is 5697.98€ with 5834.72 kg of CO2 emissions. The best
ECBO cost design is 2.29 % less than the best solution given by BB–BC.
Table 17.14 compares the results for single objective of CO2 emission functions.
The best solution reported by the ECBO is 5682.82 kg with a cost of 5913.02€. The
best ECBO CO2 design is 2.16 % less than the best solution given by BB–BC. The
percentage comparison of the solutions confirms the previous findings.
This chapter aimed to evaluate the usefulness of the ECBO and NSECBO through
the optimization of three multistory-multi-bay frames based on the ACI Code
including architectural and reinforcement detailing. The algorithm is applied to
two objective functions: the cost of material and the embedded CO2 emissions
during the construction process. Based on the present work, the following conclu-
sions can be derived:
1. The ECBO design improved the results from both objective functions in a
reasonably practical time over the designs developed by the BB–BC algorithm.
Moreover, in comparison with other evolutionary approaches, the ECBO algo-
rithm is simple to implement and it requires a few parameters to be set. These
findings proved that ECBO-based methodology could be applied as an effective
and powerful algorithm to arrive at a realistic design solution for real complex
problems.
2. Conclusive solution of the algorithm is improved through selecting more ratio-
nal groups of the elements. This implies that grouping in which the members in
the same group are similar in the internal force distribution results in more
economical solutions.
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Table 17.13 Design results for cost objective for two-bay six-story frame
17.5
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Best cost (€) 5831.70 5697.98
Average (€) 6416.73 6236.61
Std deviation (€) 219.05 369.43
CO2 emission (kg) 6306.40 5834.72
347
348
Table 17.14 Design results for CO2 objective for two-bay six-story frame
BB–BC [13] Kaveh and Ardalani [1]
Concrete Concrete
Member Group strength Width Depth strength Width Depth
type no. (MPa) (mm) (mm) Bars (MPa) (mm) (mm) Bars
Beam 1 35 230 560 1#7+1#11 50 300 500 1#9+2#8
2 30 220 550 3#4+2#8 50 290 530 3#6+3#7
3 25 250 620 4#4+4#5 45 230 450 3#7+3#7
4 25 230 550 1#8+3#6 45 250 430 2#6+3#7
5 25 230 550 1#8+1#10 40 250 490 1#9+3#6
6 25 230 550 1#8+3#6 30 260 510 3#7+3#5
17
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Best CO2 emission (kg) 5808.70 5682.82
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Average (kg) 6392.72 6134.52
Std deviation (kg) 279.59 403.39
Cost (€) 5948.81 5913.02
Cost and CO2 Emission Optimization of Reinforced. . .
References 349
References
1. Kaveh A, Ardalani S (2016) Cost and CO2 emission optimization of reinforced concrete
frames using ECBO algorithm. Asian J Civil Eng 17(6):831–858
2. Mehta PK (2002) Greening of the concrete industry for sustainable development. Concr Int
24:23–28
3. IPCC (2007) The fourth assessment report of the Intergovernmental Panel on Climate Change.
IPCC, Geneva
4. Worrell E, Price L, Martin N, Hendriks C, Meida LO (2001) Carbon dioxide emissions from
the global cement industry. Annu Rev Energy Environ 26:303–329
5. Rajeev S, Krisnamoorthy CS (1998) Genetic algorithm-based methodology for design opti-
mization of reinforced concrete frames. Comput Aided Civil Infrastruct Eng 13:63–74
6. Camp CV, Pezeshk S, Hansson H (2003) Flexural design of reinforced concrete frames using a
genetic algorithm. Struct Eng ASCE 129:105–115
7. Lee C, Ahn J (2003) Flexural design reinforced concrete frames by genetic algorithm. Struct
Eng 129:762–774
8. Paya-Zaforteza I, Yepes V, Gonzalez-Vidosa F, Hospitaler A (2008) Multiobjective optimi-
zation of concrete frames by simulated annealing. Comput Aided Civil Infrastruct Eng
23:596–610
9. Kwak HG, Kim J (2009) An integrated genetic algorithm complemented with direct search for
optimum design of RC frames. Comput Aided Des 41:490–500
10. Kaveh A, Sabzi O (2011) A comparative study of two meta-heuristic algorithms for optimum
design of reinforced concrete frames. Struct Eng 9:193–206
11. Akin A, Saka MP (2015) Harmony search algorithm based optimum detailed design of
reinforced concrete plane frames subject to ACI 318-05 provisions. Comput Struct 147:79–95
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350 17 Cost and CO2 Emission Optimization of Reinforced. . .
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Chapter 18
Construction Site Layout Planning Using
Colliding Bodies Optimization and Enhanced
Colliding Bodies Optimization
18.1 Introduction
known for its diverse applications and is widely regarded as the most difficult
problem in classical combinatorial optimization [9]. QAP problems are known as
non-polynomial hard problems (NP-hard) and because of the combinatorial com-
plexity, these cannot be solved exhaustively for reasonably sized layout problems
[5]. As an example, for n facilities, the number of possible alternatives, that is the
number of feasible configurations, is n! with larger growth than en . This is a huge
number, even for a small n. For 10 facilities, the number of possible alternatives is
already well over 3,628,000. For 15 facilities, we are already in the 12-digit
numbers. In real problems, a project with n ¼ 15 can be considered as a small
project [10].
Due to the complexity of the site layout problems, numerous techniques have
been proposed to find solutions to these problems; however, it is very difficult to
obtain an optimal one suitable for hand calculations. Thus, optimization techniques
seem to be suitable means to search for solutions of the site layout problems. The
problem can be solved using two classes of techniques: exact algorithms and
approximate algorithms. Exact algorithms such as mathematical optimization pro-
cedures were designed to find optimum solutions. But these methods could not be
adopted for large-scale projects because of the need for huge calculations and
computational efforts [11]. Therefore, they have been only successful for a single
or very limited number of facilities, as reported by Tommelein et al. [12]. Approx-
imate algorithms are categorized into two groups, heuristic and metaheuristic
algorithms, and they are developed to get the near-optimal solution in a short and
reasonable time for handling complex real-life projects. When the number of
facilities is <15, these two types of methods are able to reach an optimal solution.
However, when the number of facilities is more than 15, the problem becomes
NP-complete. For definition of NP-complete problems, the reader may refer to
Garey and Johnson [13]. As the number of facilities increases, the computational
time increases exponentially by 2n.
Since the optimal solution is not easy to obtain for large projects, researchers
have tackled the construction site layout problem (CSLP) utilizing metaheuristic
algorithms. There are many metaheuristics that can be used to address the problem
of construction site layouts (Adrian et al. [3]).
The use of artificial neural networks was investigated by Yeh [10] to improve a
predetermined site layout planning. The model minimizes a total cost function that
includes the cost of constructing a facility at the assigned location on site and the
cost of interacting with other facilities.
The Genetic Algorithm (GA) mimics the process of natural evolution and is
routinely utilized to generate useful solutions to optimization and search problems.
GA generates solutions to optimization problems using techniques inspired by
natural evolution, such as inheritance, mutation, selection, and crossover. Numer-
ous applications of GA are suggested for the facility site layout problems (Adrian
et al. [3], Cheung et al. [14], Li and Love [15, 16], Zouein et al. [17], Mawadesley
et al. [18], and Mavadesley and Al-Jibouri [19]). Li and Love [16] presented an
investigation applying the Genetic Algorithm to attain the optimal solution for
single-objective CSLP problem to accommodate facilities of unequal area in
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18.2 Construction Site Layout Planning Problem 353
Construction site layout planning problems can be modeled as a QAP in which costs
associated with the flow between facilities are linear with respect to the distance
traveled and quantity of the flow [8]. The objective of construction site layout
planning is to assign a number of predetermined facilities (n) uniquely into a
number of predetermined locations (m) where the number of locations should be
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354 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
The objective function of several models given in Table 18.1 takes the general form
(Osman et al. [20]):
n X
X n
Minimize F ¼ W ij d ij ð18:1Þ
i¼1 j¼1
where F is the objective function and n is the number of facilities and locations.
Coefficient Wij represents either the actual transportation cost per unit distance
between facilities i and j (taking into consideration the number of trips made) or a
relative proximity weight that reflects the required closeness between facilities i and
j, and dij is the distances between facilities i and j.
Table 18.1 Different kind of objective functions in the previous researches (Osman et al. [20])
No. Pseudo model of the objective function Study reference
1 To minimize the frequency of trips made by construction Li and Love [15, 16]
personnel
2 To minimize the total transportation costs of resources Cheung et al. [14] and
between facilities Tam et al. [30]
3 To minimize the cost of facility construction and the interac- Yeh [10]
tive cost between facilities
4 To minimize the total transportation costs of resources Hegazy and Elbeltag
between facilities (presented through a system of proximity [31]
weights associated with an exponential scale)
5 To minimize the total transportation costs of resources Zouein and Tommelein
between facilities and the total relocation costs (presented [32]
through a system of proximity weights and relocation weights)
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18.3 Metaheuristic Algorithms 355
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356 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
and effective in finding the best solution for NP-hard problems, are utilized for
CSLP problem.
where x0i determines the initial value vector of the ith CB. xmin and xmax are the
minimum and the maximum allowable value vectors of variables, rand is a random
number in the interval [0, 1], and n is the number of CBs.
Step 2: Defining mass
Each colliding body (CB), Xi, has a specified mass defined as
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18.3 Metaheuristic Algorithms 357
1
fitðkÞ
mk ¼ 1
, k ¼ 1, 2, . . . , n ð18:3Þ
Xn 1
i¼1 fitðiÞ
where fit(i) represents the objective function value of the ith CB and n is the number
of colliding bodies. It should be noted that larger mass values are assigned to CBs
with better objective function values.
Step 3: Creating groups
Then CB’s objective function values are arranged in an ascending order. The
sorted CBs are divided into two equal groups:
• The lower half of the CBs are stationary CBs that have lower objective
function values. These CBs are considered as good agents.
• The CBs of the upper half are moving ones. These CBs move toward the
lower ones and then the agents with upper value of each group collide
together.
Step 4: Criteria before the collision
The initial velocities of stationary CBs are equal to:
n
vi ¼ 0, i ¼ 1, 2, . . . , ð18:4Þ
2
where vi and xi are the velocity and location vector of the ith CB in this group,
respectively, and xin2 is the ith CB pair location of xi in the previous group.
Step 5: Criteria after the collision
0
After the collision, the velocity of stationary CBs (vi ) are specified by
0
miþn2 þ εmiþn2 viþn2 n
vi ¼ i ¼ 1, 2, . . . , ð18:6Þ
mi þ miþn2 2
0
Also, the velocities of moving CBs (vi ) after the collision are
0
mi εmin2 vi n n
vi ¼ i¼ þ 1, þ 2, . . . , n ð18:7Þ
mi þ min2 2 2
where ε is the coefficient of restitution (COR) that decreases linearly from unity to
zero. Thus, it is expressed as
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358 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
iter
ε¼1 ð18:8Þ
iter max
where iter and itermax are the current iteration number and the total number of
iterations for optimization process, respectively.
Step 6: Updating CBs
New locations of the CBs are evaluated using their velocities after the collision.
The new locations of stationary CBs are
0 n
xinew ¼ xi þ rand vi i ¼ 1, 2, . . . , ; ð18:9Þ
2
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18.3 Metaheuristic Algorithms 359
Is terminating No
criterion
fulfilled?
Yes
End
Step 1: Initialization
The algorithm starts with a random initial population of agents (CBs) in an
m-dimensional search space by the following formula:
where x0i determines the initial value vector of the ith CB. xmin and xmax are the
minimum and the maximum allowable values vectors of variables, rand is a random
number in the interval [0, 1], and n is the number of CBs.
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360 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
where xij is the jth variable of the ith CB. xj,min and xj,max are the lower and upper
bounds of the jth variable, respectively. In order to protect the structure of CBs,
only one dimension is changed.
Step 9: Termination criterion check
After the predefined maximum iteration number, the optimization process is
terminated. If this criterion is not satisfied, go to Step 2 for a new round of
iteration.
Flowchart of the ECBO algorithm is illustrated in Fig. 18.3.
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18.3 Metaheuristic Algorithms 361
Is terminating No
criterion fulfilled?
Yes
End
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362 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
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18.5 Case Studies of Construction Site Layout Planning 363
N Number of variables
C 1
No
End C≤N
Yes
No
C = C+1
Two case studies are selected to show the applicability and performance of the CBO
and ECBO algorithms for construction site layout optimization and their results are
compared to those of the PSO. Parameter values used in these case studies are
shown in Table 18.4. The algorithms are coded in MATLAB R2011a, and the
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364 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
This case study is a medium-sized project and is taken from Li and Love [15]. The
purpose of this problem is to find the most appropriate arrangement for placing
11 facilities into 11 predetermined locations on the site. Table 18.5 shows the
11 facilities and their corresponding index numbers.
In this case study, for the construction site layout selection, two assumptions are
made:
1. Each of the predetermined locations is capable of accommodating any of the
facilities.
2. The main gate and side gate are treated as special facilities, which have been
fixed on the predetermined locations.
The objective of this case is to minimizing the total traveling distance of site
personnel between facilities. The total travel distance is based on the formulation
of Li and Love [15] as
X
n X
n X
n X
n
Minimize TD ¼ xik xjl f ij dkl
i¼1 j¼1 l¼1 k¼1
X
n X
n ð18:13Þ
Subjected to xij ¼ 1 , xij ¼ 1
i¼1 j¼1
where n ¼ number of facilities. xik ¼ 1 when the facility i is assigned to the location
k; otherwise it is equal to 0; xjl is similarly defined. Coefficient fij is the frequency of
trips made by construction personnel between facilities i and j per day. Coefficient
dkl is the distances between the locations k and l. Therefore, TD provides the total
traveling distance made by construction personnel per day.
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18.5 Case Studies of Construction Site Layout Planning 365
Table 18.5 Facilities and their corresponding index numbers for case study 1
Index number Site facilities Note
1 Site office Not fixed
2 False work workshop Not fixed
3 Labor residence Not fixed
4 Storeroom 1 Not fixed
5 Storeroom 2 Not fixed
6 Carpentry workshop Not fixed
7 Reinforcement steel workshop Not fixed
8 Side gate Fixed to 1
9 Electrical, water, and other utilities control room Not fixed
10 Concrete batch workshop Not fixed
11 Main gate Fixed to 10
The travel distances between predetermined locations are provided in Table 18.6
(Li and Love [15]).
Trip frequencies between facilities influence the site layout planning and proximity
between predetermined site facilities. Therefore, the frequencies of the trips made
between facilities on a single day are presented in Table 18.7 (Li and Love [15]).
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366 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
Table 18.8 Comparison of the results of 50 independent runs for the first case example
Difference best– Difference best–
Algorithm Best Average Worst average solution% worst solution% STD
PSO 12,546 12,560 12,756 0.112 1.647 47.39
CBO 12,546 12,558 12,768 0.096 1.769 45.51
ECBO 12,546 12,555 12,746 0.072 1.594 32.11
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18.5 Case Studies of Construction Site Layout Planning 367
Table 18.9 A comparison between the final solution of the present work and those of the
previously reported researches
Total Best layout
Algorithms distance F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11
PSOa 12,546 9 11 5 6 7 4 3 1 2 8 10
CBOa 12,546 9 11 6 5 7 4 3 1 2 8 10
ECBOa 12,546 9 11 4 5 7 6 3 1 2 8 10
GA (Li and Love [15]) 15,090 11 5 8 7 2 9 3 1 6 4 10
ACO (Gharaie et al. [27]) 12,546 9 11 6 5 7 2 4 1 3 8 10
a
Current study
The objective function is considered as the total cost per day for transporting all
resources necessary to achieve the anticipated output. The objective function based
on Cheung et al. [14] is calculated as follows:
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368 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
Table 18.11 Four types of resources and transport costs per unit distance
Mk Resources Cost per Unit
1 Aggregate, sand, and cement/concrete 5
2 Reinforcement bars 4
3 Formwork 8
4 Completed precast units 8.5
n X
X q X
q
Minimize TC ¼ TCLMk, i, j
k¼1 i¼1 j¼1 ð18:14Þ
TCLMk, i, j ¼ MLMij CMk
MLMij ¼ FLMkij Dij
where
Dij ¼ rectangular distance between the location i and location j.
CMk ¼ cost per unit distance for resource Mk flow.
TCLMk,i,j ¼ total cost of resource Mk flow between the locations i and j.
MLMki,j ¼ distance traveled of resource Mk flow per unit time between locations
i and location j.
FLMk,i,j ¼ frequency of resource Mk flow between the locations i and j per unit time.
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18.6 Concluding Remarks 369
The flow frequency of the four types of resources between the facilities is presented
in Table 18.13.
This example was solved by carrying out 30 independent optimization runs through
1000 iterations to obtain statistically significant results by PSO, CBO, and ECBO.
Statistical results of 30 independent runs are compared in Table 18.14. As it can be
seen from Table 18.14, the average, worst, and standard deviation for ECBO are,
respectively, 92,758, 102,920, and 2733.5, which are better than those of CBO and
PSO. This indicates that ECBO not only finds a better best solution but also is more
stable. The convergence curves for the ECBO, CBO, and PSO in terms of the
number of iterations are shown in Fig. 18.6, indicating that ECBO has better
convergence rate than others. Table 18.15 summarizes the results obtained by the
present work and those of the previously reported researches. In this case study, the
best result is 92,758 which is better than that of GA, multi-searching TS, and MIP,
and it is the same as that of the Harmony search.
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370 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
Table 18.13 The flow frequency of the four types of resources between the facilities
Facility
Flow frequency 1 2 3 4 5 6 7 8 9 10 11
1. Aggregate, sand, and cement
Facility 1 20
2 15
3 35 35
4
5
6
7 20 15 35
8
9
10 35
11
2. Reinforcement
Facility 1 30
2 20
3
4 30 20 50
5
6 50 50
7
8
9
10 50
11
3. Formwork
Facility 1
2
3
4
5 48
6
7
8
9
10 48
11
4. Completed precast units
1 28
2 20
3
4
(continued)
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18.6 Concluding Remarks 371
Table 18.14 Comparing of the results of 30 independent runs for second case example
Difference best– Difference best–
Algorithm Best Average Worst average solution% worst solution% STD
PSO 92,758 97,667 106,630 5.292 14.955 3363.1
CBO 92,758 97,504 103,038 5.117 11.083 3149
ECBO 92,758 96,670 102,920 4.217 10.955 2733.5
simple formulation to find minimum of objective functions and does not depend on
any internal parameter. In order to improve the exploration capabilities of the CBO
and to prevent a premature convergence, ECBO uses a mechanism to escape from
local optimal. The latter also uses a Colliding Memory to save a number of the so
far best solutions to reduce the computational cost. To validate the models, two case
studies are considered. The results verify that the proposed approach performs very
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372 18 Construction Site Layout Planning Using Colliding Bodies Optimization and. . .
Table 18.15 A comparison between the final solution of the present work and those of the
previously reported researches
Total Best layout
Algorithms cost F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11
PSOa 92,758 5 7 9 6 1 10 8 3 11 2 4
CBOa 92,758 5 7 9 6 1 10 8 3 11 2 4
ECBOa 92,758 5 7 9 6 1 10 8 3 11 2 4
GA (Cheung et al. [14]) 99,788 1 10 9 6 8 5 11 3 7 4 2
Multi-searching TS (Liang 94,858 5 7 10 8 1 9 6 3 11 2 4
and Chao [35])
Harmony search (Kaveh 92,758 5 7 9 6 1 10 8 3 11 2 4
[33])
MIP (Wong et al. [2]) 98,424 1 10 8 6 7 5 9 3 11 4 2
a
Current study
well both in finding better results and using lower number of evaluations to find the
optimum. Comparison of the results with some other well-known metaheuristics
shows the suitability and efficiency of the utilized algorithms in CSLP. The
proposed algorithms are highly competitive with other metaheuristic algorithms
in quality of solutions and convergence speed.
References
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