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Research Article
Seismic Control of Tall Buildings Using Distributed Multiple
Tuned Mass Dampers
Hamid Radmard Rahmani and Carsten K¨
onke
Institute of Structural Mechanics, Bauhaus-Universit¨
at Weimar, Marienstr. 15, D-99423 Weimar, Germany
Correspondence should be addressed to Hamid Radmard Rahmani; radmard.rahmani@gmail.com
Received 5 March 2019; Revised 16 July 2019; Accepted 29 July 2019; Published 19 September 2019
Guest Editor: Ed´
en Boj´
orquez
Copyright ©2019 Hamid Radmard Rahmani and Carsten K¨
onke. is is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
e vibration control of tall buildings during earthquake excitations is a challenging task because of their complex seismic
behavior. is paper investigates the optimum placement and properties of the tuned mass dampers (TMDs) in tall buildings,
which are employed to control the vibrations during earthquakes. An algorithm was developed to spend a limited mass either
in a single TMD or in multiple TMDs and distribute it optimally over the height of the building. e nondominated sorting
genetic algorithm II (NSGA-II) method was improved by adding multivariant genetic operators and utilized to simulta-
neously study the optimum design parameters of the TMDs and the optimum placement. e results showed that, under
earthquake excitations with noticeable amplitude in higher modes, distributing TMDs over the height of the building is more
effective in mitigating the vibrations compared to the use of a single TMD system. From the optimization, it was observed that
the locations of the TMDs were related to the stories corresponding to the maximum modal displacements in the lower modes
and the stories corresponding to the maximum modal displacements in the modes which were highly activated by the
earthquake excitations. It was also noted that the frequency content of the earthquake has significant influence on the
optimum location of the TMDs.
1. Introduction
Given the modern development plans of large cities, which
are designed to answer the needs of their fast-growing
population, it is anticipated that the buildings in such cities
will become taller and more expensive [1]. As result, the area
of investigating solutions to provide safety and serviceability
of tall buildings in case of natural hazards such as strong
winds and earthquakes has gained much attention in the last
decade.
e current seismic design codes allow the structures to
undergo inelastic deformations during strong earthquakes.
Such structures would experience larger deformations but
less seismic forces; otherwise, the structure should sustain
much larger earthquake loads.
On the contrary, the deformations under wind and
earthquake loads are limited because of stability and ser-
viceability provisions. e resultant structures are stiff
enough to withstand the wind loads, without forming
noticeable deformations, while simultaneously being ductile
enough to withstand strong earthquakes by adopting non-
linear behaviors.
However, particularly for controlling the vibrations in
tall buildings, the code-based approaches do not necessarily
lead to an applicable and affordable solution, as these
structures need to withstand much larger wind and earth-
quake loads even though they have much lower lateral
stiffness compared to low- and midrise buildings. Moreover,
because of their very high construction costs, they are usually
designed to endure for longer time periods, which increase
their risk of experiencing strong earthquakes over the course
of their service life.
A modern answer to these issues is the idea of structural
control systems that include a variety of techniques, which
can be classified into four main categories: passive, active,
semiactive, and hybrid.
From a historical point of view, passive control systems
such as base isolations and tuned mass dampers (TMDs)
Hindawi
Advances in Civil Engineering
Volume 2019, Article ID 6480384, 19 pages
https://doi.org/10.1155/2019/6480384
were the first of these techniques to be implemented.
Considerable research has focused on the passive controller
systems, and they are already utilized in many countries
[2, 3]. As these systems need no external power supply, they
are easier to implement and design, when compared to other
advanced controllers. In structural control problems, TMDs
have been successfully implemented in different structures
such as bridges [4, 5] and buildings [6–11] to reduce
earthquake- and wind-induced vibrations. Observations of
TMDs show that they can effectively reduce vibrations in
structures that are excited by high winds, high-speed trains,
and traffic loads and also help decrease the discomfort of the
inhabitants during minor earthquakes [5, 12, 13]. Under
earthquake excitations, the literature shows that the per-
formance of the TMDs decreases as the duration of exci-
tation shortens. erefore, the TMDs are more effective for
structures subjected to narrowband long-duration far-fault
(FF) excitations compared to pulse-like near-fault (NF)
ground motions. As single TMD systems can be tuned to a
particular frequency, they are very sensitive to mistuning
and uncertainties. As a solution, multiple tuned mass
dampers (MTMDs) were first introduced by Xu et al. in 1990
[14], after which they have been studied in several research
studies [8, 9, 11, 15–19]. Li and Qu [20] considered the
structure as a single-degree-of-freedom (SDOF) system,
when connected to multiple TMDs, and studied the opti-
mum design parameters for those TMDs. With respect to
using MTMD systems in multistory buildings, Chen and Wu
[8] studied the efficiency of using multiple TMDs in miti-
gating the seismic responses in a six-story building. After
that, Sakr [15] used partial floor loads as MTMDs. ese
research studies show that the MTMD systems cover a wider
frequency range and are less sensitive to the uncertainties of
the system.
In addition to multistory buildings, several tall buildings
have benefited from the utilization of TMDs in controlling
their vibrations (see Figure 1).
As has been shown, most of them are equipped with a
single TMD, which is placed on the top level of the building.
e studies also showed that using a single TMD on the top
levels of the tall buildings can effectively reduce wind-in-
duced motions [22, 23]. is is because the structures re-
spond to the wind excitation with respect to their first
structural mode in which the top levels of the building have
maximum modal displacement. erefore, placing a single
TMD on the top level with a tuning frequency closer to the
fundamental structural frequency can efficiently reduce the
structural responses. In another research, Elias and Matsagar
[9] studied the use of distributed MTMD systems in re-
ducing wind-induced vibrations in a tall building. ey
concluded that the distributed MTMDs are more effective, as
compared to a single TMD system and an MTMD system in
which all the TMDs are placed on the top level.
e research studies show that TMDs are also effective in
mitigating earthquake vibrations in buildings. Arfiadi and
Hadi [6] used a hybrid genetic algorithm method to find the
optimum properties and the location of a TMD for a 10-
story building under earthquake excitation. In another re-
search, Pourzeinali et al. [24] utilized multiobjective
optimization to outline the design parameters of a TMD in a
12-story building under earthquake excitation. Li [11]
proposed a novel optimum criterion to optimize the
properties of double TMDs for structures under ground
acceleration. In another research, Elias et al. studied the
effectiveness of a distributed TMD system in vibration
control of a chimney [25]. e research developments in
passive control of structures using TMDs are summarized by
Elias and Matsagar [26].
In all the research studies about tall buildings cited here,
either the parameters of the TMDs have mainly been studied
under wind-induced vibrations or a single TMD has been
studied under earthquake excitation; currently, studying the
optimum parameters of TMDs under earthquake excitation,
without limiting the number and location of TMDs, is still a
challenging task because of the stochastic nature of the
earthquakes and the complex seismic behavior of such
buildings, which mandate extensive and thorough studies.
Additionally, in contrast to the wind loads, during
earthquakes, the higher modes may have more noticeable
participation in the total response of tall buildings. is is
mainly because of the (1) low frequency of the higher modes
in these structures, compared to low- and midrise buildings,
and (2) wide frequency content of the earthquakes that may
activate multiple modes in such buildings. erefore, only
controlling the lower modes by placing TMDs on the top
levels would not necessarily lead to the optimum solution for
controlling the motions in these buildings during
earthquakes.
1.1. Problem Definition. is paper addresses the mentioned
issues by studying the optimum placement and properties of
TMDs in a 76-story benchmark building as the case study
which is subjected to seven scaled earthquake excitations.
e variables of the resultant optimization problem include
the positions and properties of the TMD. e goal of the
optimization is to reduce the controlled-to-uncontrolled
ratio of the displacement, velocity, and acceleration seismic
responses. In order to solve such multiobjective optimiza-
tion problem, an improved revision of the nondominated
sorting genetic algorithm II (NSGA-II) is developed and
utilized.
In each loop, the algorithm generates an arrangement
and properties of the TMDs using the NSGA-II method and
sends them to the analyzer module to determine the re-
sponses of the building equipped with such a TMD ar-
rangement under different earthquake excitations. Based on
the responses, the algorithm assigns a fitness value for such
TMD arrangements. e fitness value is an index that shows
how good or bad the obtained responses are. e NSGA-II
then utilizes a refined history of the TMD arrangements and
corresponding fitness values for its next suggestion in the
next loop. In this study, the algorithm was allowed to spend
an applicable mass in a single TMD or distribute it through
multiple TMDs over the height of the building.
1.2. Contributions. is research incorporates several con-
tributions in the field of passive control of tall buildings and
2Advances in Civil Engineering
optimization problems. First, the issues with a single TMD
system in controlling tall buildings are addressed, and im-
provements are proposed by studying multimode control via
distribution of the TMDs over the height of the building.
Likewise, it investigates how the frequency content of the
earthquake can affect the optimum position and properties
of the TMDs. Moreover, the performance of the NSGA-II
algorithm is enhanced by adding multivariant genetic op-
erations, and the resulting algorithms are presented. Finally,
the optimum hyperparameters of the genetic algorithm for
tackling similar problems are proposed by performing
sensitivity analysis.
1.3. Outlines. e mathematical settings of a structural
dynamic problem are mentioned in Section 2. en, the
NSGA-II method is described in Section 3. After that, the
case study is presented in Section 4, and the selection and
scaling of the earthquakes are noted. en, the results of the
sensitivity analysis of the GA hyperparameters for opti-
mizing the performance of the GA algorithm are presented,
and the optimization process is then detailed in Section 7.
Finally, the obtained results are presented and discussed in
Sections 8 and 9, and the relevant conclusions are drawn.
2. Mathematical Model of the Building
e governing equation of the motion of a tall building
under earthquake excitation is as follows:
[M]{ €
U} +[C]{ _
U} +[K]U
{ } �Pt
,(1)
where M,K, and Crepresent the mass, the stiffness, and the
damping matrices of the structure and the TMDs:
[M] � Mst
+Mt
,
[C] � Cst
+Ct
,
[K] � Kst
+Kt
.
(2)
Indexes st and tindicate the degree of freedom (DOF) of
the building and the TMDs, respectively.
e external load vector, Pt, in Equation (1) comprises
inertial forces due to ground accelerations as follows:
Pt
� − €
ug[M]1t
,(3)
where 1t
(N+n)×1� [1,1,...,1]Tand the term €
ugrepresents
the ground accelerations.
e structural responses, including displacement,
velocity, and acceleration matrices, can be expressed as
follows:
U
{ } �ust1, ust2,. . . , ustN, ut1, ut2,. . . , utn
,
V
{ } �vst1, vst2,. . . , vstN, vt1, vt2,. . . , vtn
,
A
{ } �ast1, ast2,. . . , astN, at1, at2,. . . , atn
,
(4)
where Nand nrepresent the number of DOFs (DOF) for
the building and the TMDs, respectively. erefore, the
dimensions of the M,K, and Cmatrices are
(N+n) × (N+n).
150 m
300 m
450 m
600 m
Shanghai Tower
Shanghai, 2015
Building height: 632 m
Damper position:
125F/581 m
Type of damper: TMD
Ping An Finance Center
Shenzhen, 2017
Building height: 599 m
Damper position: 113F/556 m
Type of damper: TMD
Shanghai World Financial Center
Shanghai, 2008
Building height: 492 m
Damper position: 90F/394 m
Type of damper: ATMD
Petronas Twin Towers 1 and 2
Kuala Lumpur, 1998
Building height: 452 m
Damper position: within four legs under
skybridge (approx. 150 m above ground floor)
Type of damper: TMD
Princess Tower
Dubai, 2012
Building height: 413 m
Damper position: 98F/363 m (estimated)
Type of damper: TLD
23 Marina
Dubai, 2012
Building height: 392 m
Damper position: 86F/306 m
Type of damper: TMD
Almas tower
Dubai, 2008
Building height: 360 m
Damper position:
48-49F/212 m
Type of damper: TMD
TAIPEI 101
Taipei, 2004
Building height: 508 m
Damper position:
88F/378 m
Type of damper: TMD 432 park avenue
New York City, 2015
Building height: 426 m
Damper position:
85F/397 m
Type of damper: TMD
Figure 1: Tallest completed buildings with dampers [21].
Advances in Civil Engineering 3
e design parameters of a TMD include its damping,
tuning frequency, and mass. Generally, the ratios of these
parameters to the corresponding values of the structures
have more importance and are utilized in the design
procedures:
m0�mt
mst
,
β�]t
]st
,
ψ�ct
cst
,
(5)
where the parameters m0,β, and ψrefer to mass, frequency,
and damping ratios, while the indexes t and st indicate the
TMD and the structural properties.
In order to solve the equations of the motion, Newmark’s
βmethod is utilized. e average acceleration method is
considered by setting c�1/2 and β�1/4 in the related
formulations [27].
3. The Fast and Elitist Multiobjective Genetic
Algorithm NSGA-II
3.1. Introduction. In this paper, the NSGA-II method [28] is
utilized to investigate the optimum arrangement and
properties of TMDs in a tall building. NSGA-II is a non-
domination-based genetic algorithm invented for multi-
objective optimization problems. In this method, the initial
population is randomly generated, as in a normal GA
procedure, and then the algorithm sorts the population with
respect to the nondomination rank and the crowding
distance.
In general, Xdominates Y if Xis no worse than Yin all
the objectives and if Xis better than Yin at least one
objective.
Among the nondominated solutions or a union of the
first ranks of nondominated solutions, NSGA-II seeks a
broad coverage. is will be achieved by crowding distance,
which is the Manhattan distance between the left and right
neighboring solutions for two objectives.
3.2. Repair. e repair method makes infeasible solutions
feasible. Figure 2 schematically shows the repair approach
for a solution space with an infeasible solution and two
solutions in the feasible region. In this research, an infeasible
solution includes the out-of-limit properties for TMDs. As is
shown, the repair function would project each of these
infeasible solutions to the closest feasible solution. e
developed repair function calculates the shortest distance of
the TMD properties (m0,β,and ψ) in the infeasible solution
and corrects the chromosome with respect to the calculated
distance (see Algorithm 1).
3.3. Selection. e objective of selection is to choose the fitter
individuals in the population to create offsprings for the next
generation and then place them in a group commonly
known as the mating pool. e mating pool is then subjected
to further genetic operations that result in advancing the
population to the next generation and hopefully closer to the
optimal solution. In this research, the roulette wheel se-
lection method was utilized for developing the selector
function. As is also shown in Algorithm 2, the algorithm
selects the individuals based on a probability proportional to
the fitness. As is schematically illustrated in Figure 3, the
principle of roulette selection is a linear search through a
roulette wheel with the slots in the wheel weighted in
proportion to the individual’s fitness values. All the chro-
mosomes (individuals) in the population are placed on the
roulette wheel according to their fitness value [30]. In this
algorithm, a probability value is assigned to each individual
in the population. Based on these probabilities, the ranges [0,
1] are divided between the individuals so that each indi-
vidual obtains a unique range. e winning individual is
then selected by generating a random number between zero
and one and finding the individual whose range includes this
random number.
In this algorithm, the ADDRANGE function assigns a
range to each individual based on their fitness value and
their position on the wheel and the RND(1) function gen-
erates a random value between zero and one.
3.4. Optimization Variables. In this study, three variables
were defined to be optimized by the NSGA. As is sche-
matically shown in Figure 4, the variables are as follows:
Constraint boundary
Infeasible
solution space
Closest
Feasible
solution space
Fittest
Optimum
Figure 2: Repair of an infeasible solution [29].
procedure REPAIR(individual)
st �DECODER(individual)
for each tmd in st
for each property in tmd
if not property in acceptableRange
property �CLOSESTINRANGE(property)
end if
tmd �RENEW(tmd, property)
end for
newIndividual �CODER(st)
end for
return newIndividual
end procedure
A
LGORITHM
1:
Repairing individuals.
4Advances in Civil Engineering
(1) Number of TMDs
(2) Position of the TMDs ⟶story number
(3) TMDs’ properties ⟶m0,β,and ψ
e variation domain for m0,β, and ψis considered to
be in an applicable range, as shown in Table 1. e
maximum value of the total mass ratio of TMDs,
mt�n
i�1m0i, is limited to 3%, which is equal to the
considered limit for each TMD. is allows the GA al-
gorithm to either spend the allowable mass in a single
TMD or divide it among multiple TMDs and distribute
them over the height of the building. As is shown, the
damping and the frequency ratio of the TMDs are also
limited to applicable values.
3.5. Encoding. In this research, binary coding has been
considered for creating genes. erefore, the design variables
of each TMD are coded into a binary string with a constant
number of genes as shown in Figure 5. Each offspring
contains the design parameters for the TMD as follows:
m0�mt
mst
,
β�]t
]st
,
ξ�ct
cst
,
(6)
where the tand st indexes correspond to the TMD and the
structure, respectively.
ut1
ut2
K1
K2
m1
m2
C1
C2
ut
Ki
m2
Ci
utn
Kn
mn
Cn
us1
us2
usi
usN
Figure 4: Building equipped with TMDs.
37%
individual E
24% individual D
15%
individual C
14%
individual B
10%
individual A
Figure 3: Roulette wheel selection.
procedure MAKINGSELECTION(population)
fitPop �FITNESS(population)
sumFit �SUM(fitPop)
percentFit �fitPop/sumFit
rangeFitAdded �ADDRANGE(fitPop, percentFit)
randN �RND(1)
for each individual in population
r�rangeFitAdded(individual)
if randN in range r
selectedIndv �individual
exit
end if
end for
return selectedIndv
end procedure
A
LGORITHM
2:
Making individual selection.
Advances in Civil Engineering 5
After developing the genes for each TMD, the chro-
mosomes are then created by combining all genes for each
solution. As a result, each chromosome contains the coded
data of all TMDs in the building. Using this definition, the
position of each TMD is represented by the position of the
related genes in the chromosome.
3.6. Genetic Operators
3.6.1. Crossover Function. In the crossover operation, two
selected parents exchange random parts of their chromo-
some to create new offsprings. An appropriate strategy for
selecting locations of the split points and the length of the
transferred genes depends on the problem characteristics
that highly affect the performance of the algorithm and the
quality of the final results.
In this regard, different alternatives have been studied in
this research to develop an appropriate crossover function.
Examples of crossover operation forms that have been
utilized in other research studies but were not appropriate
for this research are discussed as follows:
(1) Single/k-Point Crossover: Random Points in Whole
Chromosome. In the initial steps, a completely random se-
lection of the genes for crossover has been considered as a
commonly used crossover function. In this crossover type,
after the parents are nominated by the algorithm, one or k
points in the chromosome are randomly selected, and the
new offsprings are created by splitting and combining the
parents’ chromosomes at the selected points. is process
has often resulted in producing too many meaningless and
low-quality offsprings, consequently reducing the perfor-
mance of the algorithm dramatically. Examples of mean-
ingless offsprings can include TMDs without one or more
properties (e.g., without mass or stiffness).
(2) Single/k-Point Crossover: Random Points in TMD Genes.
Preventing the production of meaningless offsprings, the
crossover function was improved in this study so that the
genes related to the TMDs in each parent could be selected
for performing a k-point crossover. Although the chance of
creating meaningless offsprings was noticeably reduced, the
results showed that the efficiency of the operator in im-
proving the results was not acceptable, as following this
process, all the genes within the considered range for a TMD
would be subjected to the same operations regardless of the
genes’ positions.
For example, the genes related to the stiffness of a TMD
in a parent were exchanged with those related to the
damping properties in another parent, which is not logical.
As a result, despite its improvements compared to the first
type, the second crossover type leads to a very low con-
vergence rate because of the production of low-quality
offsprings. In addition, one possible shortcut for reaching an
optimum solution was missed; this step involves attaching
the TMD of one parent to a story in another parent.
However, the maximum convergence rate obtained by
developing a two-variant crossover function is presented in
Algorithm 3. As is shown, in this function, in each call, one
of the two developed crossover variants would be selected
randomly. ese variants are described as follows:
(i) Variant 1: in the first variation, the crossover op-
erator acts on each of the parameters of the TMDs
separately using the k-point crossover method,
meaning that, in each call, the crossover operator
acts on the stiffness, mass, or damping of the parents
and exchanges the related properties using the
k-point crossover function. e produced offsprings
have TMDs in the same locations as their parents,
but with different properties. Investigation of the
performance of this function showed that this var-
iant improves the parameters of the TMDs re-
gardless of their positions.
(ii) Variant 2: the second crossover variation acts on the
location of TMDs in the parents. e resultant
offsprings include TMDs with the same properties as
their parents but in other stories. ese two cross-
over variations are demonstrated in Figure 6.
3.6.2. Mutation Function. In the genetic algorithm, the
mutation operator randomly changes one or multiple genes
of a parent to produce new offsprings. Generally, in the
binary coded chromosome, the following function is utilized
to change the genes:
Binary mutation(gen) � 1,if gen value �0,
0,if gen value �1.
(7)
In GA problems, the mutation function helps the
algorithm to explore the solution space more broadly and
prevents it from sticking to the local minimums. In ad-
dition, a proper mutation function improves the con-
vergence speed. In this research, in order to develop an
appropriate mutation function, different variants were
studied. It is understood that developing the mutation
function without considering the characteristics of
the problem would result in producing meaningless
11 1 111 11 11 1 1100 0 0 00 00000
m0β ξ
kt
ct
mt
TMDi:
.........
Figure 5: Binary coding the TMD’s properties.
Table 1: Parameter variation domain for TMDs.
Parameters Min. value Max. value
mt(%) — 3
m0(%) 0.2 3
β0.8 1.3
ψ5 40
6Advances in Civil Engineering
offsprings. Keeping this in mind, a two-variant mutation
function was developed, which acted on the (1) genes
related to TMD parameters and (2) group of genes related
to the location of the TMDs (See Algorithm 4).
3.7. Fitness Function. During the GA procedure, each so-
lution comprised an arrangement of TMDs with different
parameters. In order to evaluate an individual solution, three
objective functions were defined to shape the fitness func-
tion. e objectives of the optimization were taken to be the
maximum ratios of displacement, velocity, and acceleration
responses in controlled condition to their uncontrolled
values as follows:
J1�max.uC
i
uUC
i
i�1,...,N
,
J2�max.vC
i
vUC
i
i�1,...,N
,
J3�max.aC
i
aUC
i
i�1,...,N
,
(8)
where iis the story number and Nis the number of stories in
the building. e pseudocode of the developed fitness
function is presented in Algorithm 5.
4. Benchmark Building
As a case study, a 76-story, 306 m tall official building
consisting of the concrete core and concrete frames was
considered. e total mass of the building was 153,000
tonnes. e initial mathematical model of the building in-
cluded 76 transitional and 76 rotational degrees of freedom
in which the rotational degrees of freedom were then re-
moved by the static condensation method to create a 76-
degree-of-freedom model. e damping matrix of the
building was calculated by considering a 1% damping ratio
for the first five modes using Rayleigh’s approach.
e first five natural frequencies of the building were
0.16, 0.765, 1.992, 3.79, and 6.39Hz. e first five mode
shapes of the building are presented in Figure 7.
5. Ground Motion Selection
In this research, the ground motions were selected and
scaled using an intensity-based assessment procedure,
considered according to ASCE/SEI 07-10. In this regard,
seven earthquakes’ real acceleration records were selected
from the Pacific Earthquake Engineering Research Center
(PEER) NGA strong motion database [31] (see Figure 8) and
then scaled using a design response spectrum [32, 33]. e
peak ground acceleration (PGA) and other seismic pa-
rameters of the considered design response spectrum are
shown in Table 2.
e specifications of the nonscaled and scaled selected
earthquakes are presented in Tables 3 and 4. e spectra of
the nonscaled and scaled excitations are shown in Figure 9.
6. Sensitivity Analysis of Genetic
Algorithm Parameters
As the parameters of GAs are highly dependent on the
characteristics of each particular problem [34], the sensitivity
analysis on the GA parameters was performed and the op-
timum values were studied. Utilization of the obtained values
for the GA parameters resulted in improving the quality of the
solutions and the performance of the algorithm. As is shown
in Table 5, during the sensitivity analysis, the crossover and
mutation probabilities were iterated, and the GA results were
compared for the El Centro earthquake excitation.
e results were then sorted using the NSGA function,
and the pareto fronts were obtained, as shown in Table 6.
Considering the first pareto set in this table, the resultant
optimum values for crossover and mutation probabilities
were 0.7 and 0.2, respectively.
As a result, the parameters of the NSGA-II were con-
sidered as they are shown in Table 7. e sufficiency of 500
generations as the limit for the number of generations was
then evaluated, as shown in Figure 10.
7. Optimization Process
In order to study the optimum arrangement and properties
of the TMD in the benchmark building, a computer code
was developed based on the previously discussed theories
and functions. e pseudocode of the program is presented
in Algorithm 6. In order to improve the performance of the
developed code and reduce the computation time, some
advanced computer programming techniques, such as
parallel computing, were utilized. As a result, the code
utilized multiple CPU cores to produce multiple populations
in parallel, and then all the populations were combined and
sorted in each generation.
(1) procedure CROSSOVER(individual1,individual2)
(2) if rnd(1) <0.5
(3) rndTmd1 �RANDOMSELECT TMD in individual1
(4) rndTmd2 �RANDOMSELECT TMD in individual2
(5) k�rndint(3)
(6) tmd1New �K-POINTCROSSOVER(rndTmd1)
(7) tmd2New �K-POINTCROSSOVER(rndTmd2)
(8) offspring1 �REBUILD(individual1, tmd1New)
(9) offspring2 �REBUILD(individual2, tmd2New)
(10) else
(11) rndTmd1�RANDOMSELECT TMD in individual1
(12) rndTmd2�RANDOMSELECT TMD in individual2
(13) tmd1New �rndTmd2
(14) tmd2New �rndTmd1
(15) offspring1 �REBUILD(individual1, tmd1New)
(16) offspring2 �REBUILD(individual2, tmd2New)
(17) end if
(18) return (offspring1, offspring2)
(19) end procedure
A
LGORITHM
3:
Crossover operator.
Advances in Civil Engineering 7
8. Results
e results of the optimization process, including the op-
timum arrangement of the TMDs and their properties, are
shown in Figures 11–17. e drift and acceleration responses
for the top story, as the critical story, are summarized in
Table 8. As shown, the optimum number of TMDs is more
than one for some of the excitations.
e maximum number of TMDs is three, which cor-
responds to Bam and Manjil excitations, placed in 76, 75,
0000
00
00000
00000000
000
0000 00
00 0
00 000
0000 0
000
000
0
00
00
000
000011
0011 1
1
11
1
1
11111
111111
111
11 1 1
11
11 0
00
00100110011
101
11111 11
1111
1111
11111...
....
...
..
..
... ...
.........
...
... ... ...
TMD n
TMD n
Parent 1
Parent 2 TMD 20
TMD 20TMD 1
TMD 1
Offspring 1
Offspring 2
Figure 6: Crossover types.
(1) procedure MUTATION(individual)
(2) if rnd(1) <0.5
(3) rndTmd �RANDOMSELECT TMD in individual
(4) newStory �rndint(76)
(5) tmdNew �MOVETMD(rndTmd, newStory)
(6) offspring �REBUILD(individual, tmdNew)
(7) else
(8) rndTmd �RANDOMSELECT TMD in individual
(9) targetGenCount �rndint(10)
(10) for i�1to targetGenCount
(11) targetGen �RANDOMSELECTGEN(rndTmd)
(12) mutTmd �BINARYMUTATION(rndTmd, targetGen)
(13) end for
(14) offspring �REBUILD(individual, tmdNew)
(15) end if
(16) return (offspring)
(17) end procedure
A
LGORITHM
4:
Mutation operator.
(1) procedure FITNESS(benchmarkData, individualTmdAdded, excitation)
(2) ucontrolled
max , vcontrolled
max , acontrolled
max
�MAXRESPONSE(benchmarkData, individualTmdAdded, excitation)
(3) type uuncontrolled
max , vuncontrolled
max , auncontrolled
max
as constant
(4) j
1
�ucontrolled
max \uuncontrolled
max
(5) j
2
�vcontrolled
max \vuncontrolled
max
(6) j
3
�acontrolled
max \auncontrolled
max
(7) indvFitnessAdded �[individualTmdAdded, j1, j2, j3
]
(8) return (indvFitnessAdded)
(9) end procedure
A
LGORITHM
5:
Fitness of individuals.
8Advances in Civil Engineering
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0–0.5–1 0 10.5
Mode 1, f = 0.16Hz
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0–0.5–1 0 10.5
Mode 5, f = 0.76Hz
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0–0.5–1 0 10.5
Mode 3, f = 1.99Hz
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0–0.5–1 0 10.5
Mode 4, f = 3.79Hz
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0–0.5–1 0 10.5
Mode 5, f = 6.39Hz
Figure 7: Five mode shapes of the 76-story building.
Time (sec)
54 56 58 6046 48 50 5238 40 42 4430 32 34 3622 24 26 2814 16 18 20426810120
Acceleration (g)
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
(a)
Time (sec)
54 56 58 60
46 48 50 5238 40 42 4430 32 34 3622 24 26 2814 16 18 20426810120
–0.6
–0.4
–0.2
0.2
0
0.4
0.6
0.8
Acceleration (g)
(b)
Time (sec)
54 56 58 6046 48 50 5238 40 42 4430 32 34 3622 24 26 2814 16 18 20426810120
0
–0.6
–0.4
–0.2
0.2
0.4
0.6
0.8
Acceleration (g)
(c)
Figure 8: Continued.
Advances in Civil Engineering 9
Time (sec)
54 56 58 6046 48 50 5238 40 42 4430 32 34 3622 24 26 2814 16 18 20426810120
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
Acceleration (g)
(d)
604 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 5820
Time (sec)
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
Acceleration (g)
(e)
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
Acceleration (g)
246810121416182022242628303234363840424446485052545658600
Time (sec)
(f)
246810121416182022242628303234363840424446485052545658600
Time (sec)
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
Acceleration (g)
(g)
Figure 8: Earthquake acceleration records obtained from Pacific Earthquake Engineering Research Center (PEER) [31]. (a) Bam, Iran, 2003.
(b) El Centro, USA, 1940. (c) Kobe, Japan, 1995. (d) Manjil, Iran, 2002. (e) Northridge, USA, 1971. (f) Landers, USA, 1992. (g) San Fernando,
USA, 1994.
Table 2: Parameters of the design response spectrum.
Site class PGA SsS1FaFvSMS SM1SDS SD1
B 0.919 2.431 g 0.852 g 1 1 2.431 g 0.852 g 1.621 g 0.568 g
10 Advances in Civil Engineering
Table 3: Original earthquakes’ specifications.
Accelerogram Max. acceleration (g) Max. velocity
(cm/sec) Max. displacement (cm) Effective design
acceleration (g)
Predominant
period (sec)
Significant
duration (sec)
Bam 0.80 124.12 33.94 0.69 0.20 8.00
El Centro 0.44 67.01 27.89 0.30 0.06 11.46
Kobe 0.31 30.80 7.47 0.28 0.42 6.20
Manjil 0.51 42.45 14.87 0.47 0.16 28.66
Northridge 0.45 60.14 21.89 0.45 0.42 10.62
Landers 0.72 133.40 113.92 0.52 0.08 13.15
San Fernando 0.22 21.71 15.91 0.20 0.00 13.15
Table 4: Scaled earthquakes’ specifications.
Accelerogram Max. acceleration (g) Max. velocity (cm/sec) Max. displacement (cm) Effective design
acceleration (g)
Predominant
period (sec)
Significant
duration (sec)
Bam 0.78 128.55 33.95 0.64 0.20 8.24
El Centro 0.57 75.76 28.36 0.48 0.32 9.16
Kobe 0.56 38.83 15.68 0.56 0.36 4.16
Manjil 0.68 42.06 15.02 0.55 0.08 28.26
Northridge 0.59 64.57 22.16 0.58 0.40 10.40
Landers 0.73 141.02 113.78 0.56 0.08 12.92
San Fernando 0.60 23.71 15.88 0.61 0.10 6.03
0 0.5 1 1.5
0.5
1
1.5
2 2.5 3 3.5 4
Period (s)
Acceleration (g)
Mean matched spectrum
Target spectrum
Figure 9: Mean matched spectrum compared to the target spectrum.
Table 5: Crossover and mutation variations.
Variation no. Crossover Mutation J
1
J
2
J
3
1
0.6
0.1 0.845 0.917 0.941
2 0.2 0.860 0.926 0.950
3 0.3 0.855 0.924 0.948
4 0.4 0.839 0.915 0.942
5
0.7
0.1 0.862 0.925 0.946
6 0.2 0.835 0.913 0.939
7 0.3 0.861 0.925 0.947
8 0.4 0.846 0.919 0.944
9
0.8
0.1 0.869 0.929 0.950
10 0.2 0.876 0.932 0.952
11 0.3 0.842 0.915 0.941
12 0.4 0.852 0.922 0.947
13
0.9
0.1 0.871 0.930 0.951
14 0.2 0.866 0.927 0.947
15 0.3 0.874 0.932 0.953
16 0.4 0.839 0.913 0.939
Advances in Civil Engineering 11
and 64th floors in both cases. Under both sets of excitations,
most of the mass for the TMDs was dedicated to those on the
top two floors. e tuned frequency of the TMDs was close
to the fundamental frequency of the building for the top two
TMDs and about 1.24 times the fundamental frequency for
the TMD in the 64th-floor building. e damping ratio of
the TMDs placed on the top two stories was close to the
maximum allowed value, which was 40, while it was about 14
for the TMD in the 64th-floor building.
It was observed that the controlled displacement re-
sponses of the building improved substantially by about 65%
under the Manjil earthquake excitation. On the contrary, the
objective J
1
for the Bam earthquake had a value of about 0.95,
which means to about 5% improvement in reducing max-
imum displacements, compared to the uncontrolled re-
sponse. is low objective value was also obtained under the
Landers earthquake, with a value of about 0.93 for J
1
, im-
plying 7% improvement in reducing the displacement
responses. However, the displacement response shows
substantial improvements in damping further oscillations
compared to uncontrolled buildings.
For Landers, Northridge, and San Fernando earth-
quakes, the TMDs were placed on stories 76 and 74, and
most of the allowed mass was dedicated to the TMD on the
roof. e optimum tuning frequency of the TMDs was close
to the fundamental frequency of the building. e damping
values of the TMDs were between 36.39 and 39.78, which
were close to the maximum considered damping ratio.
For the El Centro and Kobe earthquakes, the optimum
results were obtained by placing a single TMD system on the
roof. In both cases, all the allowed mass was utilized in the
TMD. Under these excitations, the frequency ratios of the
TMDs were registered as 1.06 and 1.05, which indicated a
tuning frequency closer to the fundamental frequency of the
building; the damping ratios of the TMDs were 39.53 and
39.78, which were close to the maximum allowed value.
9. Discussion
As mentioned in Results, the optimum target stories for
placing the TMDs included the top two stories for all of the
earthquake excitations and some other stories such as 74 and
64 for some of the earthquakes. In order to understand the
reasons behind this optimum arrangement, the building’s
mode shapes are again presented in Figure 18, but in each
mode, the stories with maximum displacements are also
marked. As shown, for the first three modes, the top three
stories have the maximum displacements. e roof and the
75th and 61st stories, for the 4th mode, and the roof and the
75th and 64th stories, for the 5th mode, were the stories with
maximum modal displacements.
Consequently, it can be concluded that placing a TMD
on the top stories would improve the modal displacements
in all five modes, an observation which agrees with the
optimization results.
On the contrary, for some earthquakes, TMDs were
placed in the lower stories, which implies that the optimum
placement of the TMDs may also be related to some exci-
tation parameters. For this reason, fast Fourier trans-
formation (FFT) was performed for each earthquake’s
excitation record, and amplitudes for each building’s mode
frequency were then specified to investigate the effective
properties of the excitations, as shown in Figure 19.
As shown here, unlike other earthquakes, for the Bam
and Manjil earthquakes, the amplitude of the excitation in
the 4th and 5th modes is more than that in the lower modes.
As a result, although these higher modes have lower mass
participation factors, their participation in the total response
of the earthquake is increased by higher excitation ampli-
tudes. In order to theoretically study these results, the
displacement response of the building under the ground
motion is presented in equation (9) as sum of the modal
nodal displacements:
u(t) �
N
n�1
un(t),(9)
Table 6: NSGA of crossover and mutation variations.
Pareto front Crossover variation no. Mutation variation no.
1 6 —
2 16 —
3 4 11
4 1 —
5 8 —
6 5 12
7 3 7
8 2 14
9 9 —
10 13 —
11 10 15
Table 7: Initial parameters of NSGA-II.
Number of
generations
Population
size
Crossover
probability
Mutation
probability
500 100 0.7 0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
50 100 150 200 250 300 350 400 450 500
Generation
J1
San Fernando
Northridge
Landers
El Centro
Bam
Manjil
Kobe
Figure 10: Trend of objectives during the generations.
12 Advances in Civil Engineering
where unrepresents the nth mode’s displacements. e
contribution of the nth mode to the nodal displacement u(t)
is
un(t) � ϕnqn(t),(10)
where qnrefers to the modal coordinate which can be
calculated from the following equation:
€
qn+2ζnωn_
qn+ω2
nqn� − Γn€
ug(t),(11)
where Γnis the modal participation factor of the nth mode
and is the degree to which the nth mode participates in the
total response. e modal participation factor can be cal-
culated based on the modal displacements and masses as
follows:
(1) population �INITIALIZEPOPULATION()
(2) repeat
(3) repeat
(4) {parent1, parent2} �MAKESELECTION (population)
(5) if rnd(1) ≥crossoverProbability
(6) {offspring1, offspring2} �CROSSOVER(parent1, parent2)
(7) REPAIR(offsprint1, offspring2)
(8) COMPUTEFITNESS(offsprint1, offspring2)
(9) end if
(10) if rnd(1) ≥mutationProbability
(11) {offspring1, offspring2} �MUTATION(parent1, parent2)
(12) REPAIR(offsprint1, offspring2)
(13) COMPUTEFITNESS(offsprint1, offspring2)
(14) until size(population) ≤M
(15) tempPopulation �population
(16) newPopulation �[ ]
(17) repeat
(18) p�PARETOFRONT(tempPopulation)
(19) ps �CROWDINGDISTANCE(p)
(20) newPopulation �newPopulation + ps
(21) tempPopulation �tempPopulation-ps
(22) until size(tempPopulation) >2
(23) population �newPopulation
(24) until generationNumber ≤N
A
LGORITHM
6:
NSGA-II.
Uncontrolled
Controlled
10 20 30 40 50 600
Time (s)
–1.5
–1
–0.5
0
0.5
1
1.5
Displacement (m)
(a)
Story m0ß
39.53
39.46
18.37
76
75
64
1.43
0.93
0.64
0.98
1.01
1.24
ψ
(b)
J3
J1J2
0.95 0.88 0.83
(c)
Figure 11: Uncontrolled/controlled responses under the Bam earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
Advances in Civil Engineering 13
ΓN�ϕn
[M]1
{ }
ϕn
T[M]ϕn
�N
j�1mjϕjn
N
j�1mn
jϕ2
jn
.(12)
Equation (11) is related to a single-degree-of-freedom
(SDOF) system with frequency and damping corresponding
to the nth mode.
As shown here, in each mode n, the nodal displacement
un(t) has a direct relationship with the modal displacements,
ϕn, which means that the stories with maximum modal dis-
placements would have greater participation in the building’s
modal response. In addition, it is obvious that the modal
response in the nth mode is also related to the frequency
content of the earthquake excitation, €
ug, which means that a
larger acceleration amplitude at that mode’s frequency would
result in larger modal responses for that mode.
erefore, the participation of a particular story in the
total response of the building would be more than that of the
other stories if the following conditions are met:
(1) e story has maximum modal displacement in the
modes with a larger modal participation factor
(2) e story has maximum modal displacement in the
nth mode with a lower participation factor, but the
ground motion has a larger Fourier transformation
amplitude in the nth mode’s frequency (fn)
10 20 30 40 50 600
Time (s)
Uncontrolled
Controlled
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
Displacement (m)
(a)
Story m0β
39.5376 2.99 1.06
ψ
(b)
J3
J1J2
0.84 0.91 0.93
(c)
Figure 12: Uncontrolled/controlled responses under the El Centro earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
10 20 30 40 50 600
Time (s)
Uncontrolled
Controlled
Displacement (m)
–0.5
–0.4
–0.3
–0.2
0
0.1
0.2
0.3
0.4
(a)
Story m0β
39.78
76 3.00 1.05
ψ
(b)
J3
J1J2
0.86 0.90 0.97
(c)
Figure 13: Uncontrolled/controlled responses under the Kobe earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
14 Advances in Civil Engineering
ese derivations validate the possibility of placing the
TMDs in stories other than the top stories, a conclusion that
agrees with the optimization results.
10. Conclusion
Considering the obtained results and related discussions in
previous sections, the following conclusions can be drawn:
(1) Compared to a single TMD on the roof level, a
distributed MTMD system is more efficient in im-
proving structural responses with the same amount
of masses under excitation for earthquakes that have
noticeable amplitude at the structure’s frequencies at
higher modes, a scenario likely to happen within the
lifetime of a tall building.
(2) e optimum stories for placement of the TMDs
includes
(a) e stories with maximum modal displacements
in the lower structural modes
(b) e stories with maximum modal displace-
ments in modes with frequencies at which the
earthquake excitation has noticeable amplitudes
10 20 30 40 50 600
Time (s)
Uncontrolled
Controlled
Displacement (m)
–3
–2
–1
0
1
2
3
4
(a)
Story m0β
37.0776 2.01
1.00 1.01 37.13
1.01
ψ
74
(b)
J3
J1J2
0.93 0.89 0.98
(c)
Figure 14: Uncontrolled/controlled responses under the Landers earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
10 20 30 40 50 600
Time (s)
Uncontrolled
Controlled
Displacement (m)
–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
(a)
Story m0βψ
76 1.50 1.02 35.12
31.34
14.92
1.01
1.21
0.9575
64 0.55
(b)
J3
J1J2
0.36 0.57 0.95
(c)
Figure 15: Uncontrolled/controlled responses under the Manjil earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
Advances in Civil Engineering 15
10 20 30 40 50 600
Time (s)
–1.5
–1
0
–0.5
0.5
1.5
1
Displacement (m)
Uncontrolled
Controlled
(a)
Story m0β
37.0576 2.26
0.75 1.04 36.98
1.04
ψ
74
(b)
J3
J1J2
0.81 0.84 0.95
(c)
Figure 16: Uncontrolled/controlled responses under the Northridge earthquake. (a) Roof displacement response. (b) TMD specifications.
(c) Objective values.
10 20 30 40 50 600
Time (s)
Displacement (m)
Uncontrolled
Controlled
–0.8
–0.6
–0.4
–0.2
0
0.4
0.2
0.6
0.8
(a)
Story m0β
36.2776 2.28
0.72 1.03 36.39
1.03
ψ
74
(b)
J3
J1J2
0.80 0.84 0.97
(c)
Figure 17: Uncontrolled/controlled responses under the San Fernando earthquake. (a) Roof displacement response. (b) TMD specifi-
cations. (c) Objective values.
Table 8: Comparison between controlled and uncontrolled drifts and absolute accelerations of the top story for different earthquake
excitations.
Earthquake Max. drift (uncontrolled) Max. drift (controlled) Max. absolute acceleration
(uncontrolled, m/s2)
Max. absolute acceleration
(controlled, m/s2)
Bam 0.0184 0.0172 34.94 17.32
El Centro 0.0128 0.0111 20.99 16.11
Kobe 0.0079 0.0066 25.72 21.40
Manjil 0.0159 0.0074 28.51 9.30
Northridge 0.0178 0.0146 23.87 15.55
Landers 0.0282 0.0262 28.94 14.79
San Fernando 0.0092 0.0075 21.61 12.35
16 Advances in Civil Engineering
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
–1 –0.5 0 0.5 1
Mode 1, f = 0.159 Hz
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
–1 –0.5 0 0.5 1
Mode 1, f = 0.159 Hz
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
–1 –0.5 0 0.5 1
Mode 1, f = 0.159 Hz
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
–1 –0.5 0 0.5 1
Mode 1, f = 0.159 Hz
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
–1 –0.5 0 0.5 1
Mode 5, f = 6.39 Hz
Figure 18: ree stories with maximum modal displacements in the first five natural modes.
0
0.05
0.1
0.15
0.2
0.25
0.3
Amplitude
10–2 10–1 100101102
0.03
0.17
0.13
0.16
0.11
0.16
0.76
1.99
3.78
6.39
Frequency (Hz)
(a)
0.03
0.16
0.03
0.76 0.05
6.39
0.11
3.78
0.18
1.99
10–2 10–1 100101102
Frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
Amplitude
(b)
0.16
10–2 10–1 100101102
Frequency (Hz)
0.15
0.11
0.76
1.99 6.39
3.78
0.04
0.07
0.77
0
0.05
0.1
0.15
0.2
0.25
Amplitude
(c)
10–2 10–1 100101102
Frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
0.3
Amplitude
0.16
1.99
6.39
0.09
0.76 3.78
0.05
0.05
0.12
0.17
(d)
Figure 19: Continued.
Advances in Civil Engineering 17
(3) e optimum parameters for the TMDs that control
the vibrations in the lower modes include the
maximum allowed damping ratio. is indicates that
increasing the damping ratio would improve the
performance of such TMDs.
(4) e results showed that the performance of the
TMDs was not good in reducing the initial maxi-
mums in displacement responses compared to their
reduction of the later maximums that occurred after
some initial oscillations.
(5) Even in the cases with immediate maximum dis-
placement responses, the MTMD system signifi-
cantly improves the damping.
Data Availability
e data used to support the findings of this study are
available from the corresponding author upon request.
Conflicts of Interest
e authors declare that they have no conflicts of interest.
References
[1] M. M. Ali and K. S. Moon, “Structural developments in tall
buildings: current trends and future prospects,” Architectural
Science Review, vol. 50, no. 3, pp. 205–223, 2007.
[2] B. F. Spencer and M. K. Sain, “Controlling buildings: a new
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10–2 10–1 100101102
Frequency (Hz)
0
0.05
0.1
0.15
0.2
0.25
0.3
Amplitude
0.16
0.76
1.99
3.78
6.39
0.11
0.17
0.21
0.18
0.2
(e)
10–2 10–1 100101102
Frequency (Hz)
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0.11 0.1
0.03
0.16
0.76
1.99 3.78
6.39
0
0.05
0.1
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0.3
0.35
(f)
10–2 10–1 100101102
Frequency (Hz)
0
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0.25
Amplitude
0.04
0.1
0.14
0.05
0.05
0.16
0.76
1.99
3.78
6.39
(g)
Figure 19: Fourier transform of earthquake excitations. (a) Bam. (b) El Centro. (c) Kobe. (d) Landers. (e) Manjil. (f ) Northridge. (g) San Fernando.
18 Advances in Civil Engineering
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