Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering
journal homepage: www.elsevier.com/locate/soildyn
Design strategies of viscous dampers for seismic protection of building
structures: A review
T
⁎
D. De Domenicoa, , G. Ricciardia, I. Takewakib
a
b
Department of Engineering, University of Messina, 98166 Messina, Italy
Department of Architecture and Architectural Engineering, Kyoto University, Nishikyo, Kyoto 615-8540, Japan
A R T I C LE I N FO
A B S T R A C T
Keywords:
Fluid viscous dampers
Optimal damper design
Energy dissipation devices
Earthquake protection
Stochastic earthquake excitation
Numerical optimization
Fluid viscous dampers (FVDs) are well-established supplemental energy dissipation devices that have been
widely used for earthquake protection of structures. Optimal design, placement and sizing of FVDs have been
extensively investigated in the last four decades. In this review paper, an overview of the most popular methodologies from the abundant literature in the field is presented. Key aspects and main characteristics of the
different strategies to identify the optimal damping coefficients and the optimal placement of FVDs are scrutinized in a comparative manner. The optimal design problem is often solved through a numerical approach to a
constrained optimization problem, by minimizing some performance criteria that are representative measures of
the system response. With reference to two simple benchmark six-story shear-type structures subject to both a
stochastic earthquake excitation and 44 natural ground motions extracted from the FEMA P695 record set,
comparison of the seismic performance is carried out considering FVDs designed according to different methods
— an overall number of 138 different design scenarios are incorporated in this comparative study. These
methods are based either on a desired (target) damping ratio constraint or on a fixed total cost, here roughly
related to the sum of the damping coefficients of the added FVDs. Some energy-based perspectives are also given
in this review paper in order to interpret the seismic performance in terms of the amount of energy dissipated by
the FVDs, out of the total input energy from the earthquake excitation.
1. Introduction
In the field of passive energy dissipation systems for civil engineering structures [1], the use of fluid viscous dampers (FVDs) [2,3]
has gained popularity in the last few decades, mainly due to: 1) the
capability of enhancing earthquake performance through significant
energy dissipation; 2) the capability of generating forces that are out of
phase with displacements; 3) the possibility of increasing the damping
ratio of a structure without significantly altering the inherent stiffness
characteristics (which avoids typical trial-and-error repetitive design
strategies necessary for other kinds of devices like viscoelastic dampers,
hysteretic dampers or base isolators [4–6]). Motivated by the good
performance observed in aerospace and military applications, FVDs
have been increasingly implemented in structural applications in the
last four decades as supplemental energy dissipation devices for both
new and existing civil engineering structures, see e.g. [7–14] and references therein for just a few emblematic examples. Other characteristics that have contributed to the success of this technology are related
to the little sensitivity over a broad range of frequencies and
⁎
temperatures, and the reasonably compact shape if compared to the
obtainable strokes and forces [1]. Of particular relevance to the present
paper, analytical modeling of these devices can be extremely simplified
in the case of linear FVDs, i.e., considering a purely viscous behavior
underlying a simplified linear force-velocity relationship. This linear
relation is not really observed in experiments [15,16], but a fractional
velocity power-law has been found to be more suitable for fitting experimental findings. Nonetheless, the concept of “energy-equivalent”
dampers can be invoked to compromise between the actual nonlinear
(power-law) behavior and a simplified (equivalent, in terms of energy
dissipation) linear modeling [17].
A plethora of studies were carried out by different researchers to
optimize the FVDs seismic performance when implemented in civil
engineering structures, especially buildings. Pioneering works date
back to the 80 s and to the first 90 s with simple yet effective procedures
[18–21]; benefitting from more recent computer developments and new
computational algorithms, more sophisticated and refined procedures
were proposed over the following years by several researchers [22–40],
and significant advances were achieved in recent years with
Corresponding author.
E-mail address: dario.dedomenico@unime.it (D. De Domenico).
https://doi.org/10.1016/j.soildyn.2018.12.024
Received 4 October 2018; Received in revised form 26 November 2018; Accepted 20 December 2018
0267-7261/ © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 1. Design strategies of fluid viscous dampers: sketch adopted in the organization of the present review paper.
these concepts, a numerical procedure is employed to determine the
optimal set of dampers that achieve the best energy dissipation behavior (i.e., that maximize the energy dissipation) corresponding to a
desired (target) damping ratio of the fundamental mode of the building,
or corresponding to a fixed total cost somehow related to the sum of the
damping coefficients of the added FVDs, cd tot = ∑j cdj . For two simple
benchmark six-story shear-type structures under a stochastic excitation
and under 44 natural ground motions extracted from the FEMA P695
record set, comparison of the seismic performance is carried out considering FVDs designed according to different methods — an overall
number of 138 different design scenarios are incorporated in this
comparative study.
increasingly sophisticated techniques [41–55]. A summary of a few of
the quoted methodologies, at least of those developed up to the end of
the last decade, can be found in the monograph by Takewaki [56].
Other relevant contributions are collected in the book by Lagaros et al.
[57], which is focused on design optimization of active and passive
structural control systems, including viscous dampers. Even in the last
few years further methodologies or refinements of previous methods
have been developed [58–67]. The wealth of literature in the field up to
date [68–82] confirms that this topic is still of great interest and, thus,
is worth of further investigation.
This paper presents a general yet concise overview of the fundamentals of the most popular methodologies proposed in the literature so
far. These methodologies and design philosophies are different from
one another in terms of earthquake ground motion representation,
adopted structural model, addressed performance criterion, underlying
theoretical principles, assumed numerical algorithm or because of different constraint conditions in the numerical optimization problem. A
sketch of the organization proposed in the present review paper is illustrated in Fig. 1, wherein red texts and numbers indicate the section
and subsection labels. A broad variety of analytical, heuristic, and numerical methodologies developed for the design, placement and sizing
of FVDs are reviewed in order to scrutinize key aspects and main
characteristics. As a complementary contribution of this paper, some of
the reviewed methodologies are re-interpreted from energy-based perspectives. In particular, modeling the earthquake ground motion acceleration as a stochastic excitation, energy balance equations in stochastic terms are set up and used to assess the seismic performance of
different distributions of FVDs proposed in the literature. Building on
2. Structural model: building with fluid viscous dampers
2.1. Equations of motion
According to Fig. 1, the overview of the different design strategies
starts with some remarks on the structural model. For simplicity, to
establish the nomenclature adopted in the remainder of the paper, let us
consider a planar n-story building as sketched in Fig. 2, which is subject
to the horizontal ground-motion acceleration u¨ g (t ) . A set of FVDs are
installed along the building height as supplemental energy dissipation
devices. The position of the FVDs in Fig. 2 resembles a typical “interstory scheme”, i.e., the damper forces (at the j th story) are proportional to the interstory velocity. In particular, adopting a simplified
(Newtonian) linear model, the damper force at the j th story is
145
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 2. Sketch of an n-story building structure with fluid viscous damper devices.
fdj = cdj Δw˙ j = cdj f j Δu˙ j
above-written equations of motion (2) have general meaning and can
also describe the dynamics of adjacent building structures with interconnected dampers [53,87–89] for the mitigation of seismic pounding
risk [90,91], or with dampers installed against infinitely stiff reaction
towers [81] or connected to the ground according to the fixed-point
placement [41,51], the only difference being related to the definition of
the transformation matrix R for the specific problem investigated. For
interstory dampers implemented at every story through generic brace
configurations as per Fig. 2, the transformation matrix R is
(1)
where Δw˙ j = f j Δu˙ j denotes the relative velocity at the ends of the j th
damper, Δu˙ j = u˙ j − u˙ j − 1 is the interstory velocity at the j th story, and f j
is a displacement magnification factor, related to the kind of brace
supporting the damper itself [83,8,84]. Values of f j for some common
brace configurations are depicted in Fig. 2. In addition to these wellknown brace configurations, it is worth mentioning also the scissorjack-damper system, developed in [85] as a more compact and more
effective variant of the toggle-brace-damper system.
Trombetti and Silvestri [41,42,51,52] demonstrated that such interstory installation scheme is not the optimal one for enhancing the
seismic performance and maximizing the damping ratio. These authors
proposed an alternative, more effective “fixed-point” damper implementation scheme, which can be physically achieved by connecting
all dampers either to the ground or to an infinitely stiff vertical reaction
tower. These alternative layouts will not be dealt with in the sequel of
the paper, whereas attention is focused on the more widely adopted
interstory installation scheme shown in Fig. 2. It is meant that a preliminary static condensation method [86] has already been applied to
the structure in order to eliminate the (zero-mass) rotational degrees of
freedom (DOFs), so as to obtain the schematic model of Fig. 2. With
axial deformations in structural elements neglected, the mass can be
considered lumped at the floor level and the equations of motion of this
n-DOF system are expressed in matrix-vector compact form as
Mu
¨ + Cu˙ + Ku + Fd = −Mτu¨ g (t )
0 …
⎡ f1
⎢− f2 f2 0 …
R=⎢ 0 −f f 0 …
⎢
3
3
…
⎢ …
0
0
fn
…
−
⎢
⎣
Mu
¨ + (C + Cd) u˙ + Ku = −Mτu¨ g (t )
RT DR
(6)
denoting the n × n damping matrix of the added FVDs
with Cd =
and D = diag{cdj} a nd × nd diagonal matrix containing the damping
coefficients of the FVDs.
The above equations of motion (6) refer to planar buildings with
FVDs. In fact, most of the literature papers quoted above dealt with
planar building models for the sake of simplicity. For some interesting
effects on 3D frames, reference could be made to the studies reported in
[33,62,31,50,93,70,92,81], to quote just a few.
For the sake of completeness, it is worth noting that other types of
passive protective systems exploiting FVDs have been devised in the last
two decades, which are not limited to the basic technology of dissipative braces sketched in Fig. 2. Although not treated further in the
sequel of the paper, they are briefly referenced here in view of their
significant role in the field of FVD-based seismic protection strategies.
The most important technology is the “damped cable system” (as
termed by some authors [94–96]) or “supplemental tendon system” (as
named by other authors [97,98]) with some minor differences between
the two technologies. This system consists of prestressed steel cables
having sliding contacts with the floor slabs, linked to FVDs that are
in which M, C, K are the n-dimensional mass, (inherent) damping and
stiffness matrices of the building structure, respectively, τ is the influence vector of order n (in this specific case, with each element equal to
unity), while Fd is the n × 1 vector of the damper forces. In the most
general case, these forces may be expressed as
(3)
where f d is a nd × 1 vector collecting terms fdj = cdj Δw˙ j as per Eq. (1),
and R is a nd × n transformation matrix (with nd denoting the number
of dampers) expressing the relation between the vector of relative
displacements at the ends of each damper Δw and the system vector
displacement u according to
(4)
Δw = Ru.
(5)
whereas for a simpler K-brace chevron configuration at all stories
( fi = 1 ∀ i = 1, …, n ), R is a tridiagonal matrix of 1, − 1, and 0 terms,
which is termed R̃ in the sequel of the paper. Note also that the in˜ , regardless of the
terstory displacements can be expressed as Δu = Ru
brace configuration adopted for the FVDs. Substituting (1), (3) and (4)
into (2) yields
(2)
Fd = RT f d,
0⎤
0⎥
⎥
0 ⎥,
⎥
fn ⎥
⎦
j th
component of Δw in (4) is
For instance, considering Eq. (1), the
Δwj = f j (uj − uj − 1) . With this transformation matrix introduced, the
146
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
fixed to the foundation at their lower ends, and to top building floor (or
an upper floor) at their upper ends. Owing to this peculiar configuration, the FVDs dissipate energy due to the total horizontal displacement
of the building structure, in contrast to interstory FVDs installed in
dissipative bracing systems that are instead engaged by interstory drifts.
This peculiarity allows incorporating only a single FVD per vertical
alignment conveniently.
elements, especially for exciting frequencies below a cut-off frequency
of around 4 Hz [132].
Additionally, the simplification of infinitely stiff supporting braces
is quite popular in the literature, and it is also adopted in the numerical
examples of this paper. This simplification is acceptable in the framework of a preliminary design of FVDs. However, it is not strictly realistic. The flexibility of the brace might affect the damper-brace mechanical behavior in certain cases [133]. As an example, locking
phenomena of the viscous damper may occur if the damping coefficient
is very high, and this may cause concentration of deformation on the
brace. What is of critical importance is thus the ratio between FVD and
its supporting brace. Singh et al. [128] demonstrated that ignoring the
finite stiffness of the supporting braces leads to non conservative results
because it implies an overestimation of the dampers’ efficacy. This is
also confirmed by Park et al. [134] in the context of viscoelastic dampers.
Other
relevant
studies
from
the
literature
are
[29,67,64,70,135,136,129]. In this context, a recent work by Pollini
et al. [92] addressed the simultaneous optimal design of FVDs with a
sophisticated model for the damper-brace behavior. In particular, two
springs (one accounting for the damper stiffness and the other accounting for the stiffness of the supporting brace) in series with a
nonlinear dashpot (with fractional power law) were considered.
Through an iterative procedure under harmonic displacement history, it
was possible to pre-compute a ratio between the damping coefficient of
the dashpot cd and the equivalent stiffness of the brace-damper system
keq , so that the only design variable remains cd [92].
2.2. Damper nonlinearity
The actual power-law nonlinear constitutive behavior was explicitly
incorporated in some studies of the literature dealing with fluid viscous
dampers [92,99–108]. Lin et al. [109] proposed a direct displacementbased design procedure for buildings equipped with passive energy
dissipation devices, including FVDs, based on a rational linear iteration
method starting from a target displacement, from which the corresponding design force, strength and stiffness can be determined. This
procedure was later extended to moment resisting frames with viscous
dampers by Sullivan and Lago [110]. Direct, simplified methodologies
[111,112], multiple-step approaches [113], including an efficient fivestep procedure [114–117], or stochastic-based procedures [118–121]
have been proposed for the evaluation of the seismic response or for a
preliminary seismic design of structures with nonlinear fluid viscous
dampers. In this paper, attention is restricted to the linear modeling
assumption as only linear fluid viscous dampers are considered as per
Eq. (1). Various linearization procedures are available in the literature
to deal with nonlinear dampers that can be adopted at least in a preliminary design stage [17,122–124]. Nonetheless, the general concepts
underlying most of the methodologies discussed here also apply to
nonlinear fluid viscous dampers with appropriate modifications.
3. Earthquake representation
3.1. Single record versus ensemble of records
2.3. Linear or nonlinear parent frame
The design problem of FVDs depends, among many factors, also on
how the earthquake ground motion acceleration üg entering the equations of motion (6) is considered. Various options exist in this regard,
which were explored in the literature. In this context, Lavan and Levy
[48,49] were the first to develop and employ a methodology accounting
for an ensemble of realistic ground motion records with a specified
target performance index for the design problem. This approach has
obvious advantages in comparison with earlier strategies that considered each time a single record [21,37,38]. In particular, the approach proposed by Lavan and Levy [48,49] selects one specific, socalled “active” ground motion within the given ensemble (having the
maximal input energy spectrum [137] for the fundamental period of the
structure), and solves the optimization problem for that record. Then, if
the optimal solution (found for the above-mentioned active ground
motion) violates constraints for other records of the original ensemble,
a new candidate ground motion (among the remaining records) is
considered active (selected as the one associated with the largest
maximum drift). Thus, this strategy is practical and efficient as the
optimization scheme is likely to use only a few of the records (the most
critical ones) and not the whole ensemble, with a reduction of computational effort.
As an alternative to records of natural ground motions, other works
incorporated the stochastic nature of the seismic input through a power
spectral density (PSD) function, as described in the following subsection.
The majority of the studies mentioned above assumed that the
parent frame has a linear elastic behavior. In fact, the equation (6)
assumed a linear restoring force for the frame. This is a simplified
idealization that can be considered acceptable under the hypothesis of
regular structure, FVDs being fully engaged to dissipate a large amount
of earthquake input energy and moderate earthquake excitations. Also,
this assumption can be justified in view of limitations of inter-story
drifts to the values of drift for which yielding occurs [92]. On the other
hand, under sever earthquake ground motions the behavior of the
parent frame is likely to undergo plastic deformations. Additionally,
damage concentration in the presence of strength irregularities in nonregular-in-elevation buildings is expected to occur. Some studies from
the relevant literature that also incorporated the nonlinearity of the
frame response into the design of FVDs, by assuming a yielding structure or considering buildings experiencing inelastic deformations, are,
among others, those by Attard [125], Lavan and Levy [48], Lavan et al.
[126], Lavan and Dargush [58] and Cimellaro et al. [127].
2.4. Damper stiffness and stiffness of supporting braces
In this paper, the dampers are modeled as pure viscous linear
dashpot elements. In reality, the compressibility of the fluid flowing
between the chambers of FVDs is not infinite, thus leading to a finite
axial stiffness of the device. Consequently, in place of a pure viscous
dashpot, a Maxwell model composed by a linear spring in series with a
nonlinear dashpot system would be more suitable for modeling purposes [2,128,129]. Nevertheless, in practical applications and for
common fluids (silicone oils) adopted, the axial stiffness is very high,
around one order of magnitude higher than the ratio between maximum force and maximum stroke. Under these conditions, the oil can be
considered incompressible. Previous numerical and experimental works
[27,130,114,131] demonstrated the little influence of the damper
stiffness on the results, which justifies the assumption of pure dashpot
3.2. Stochastic description
In order to incorporate the stochastic nature of the seismic input,
the base acceleration üg entering the equations of motion (6) can
modeled as a random process [138]. The widely used Kanai-Tajimi
filtered Gaussian white-noise process is usually adopted for describing
the earthquake frequency content, with a second filter in series as
proposed by Clough and Penzien [139] in order to avoid the unrealistically high values of the power-spectral-density (PSD) function in
147
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
the low-frequency regime. In this model, the bedrock Gaussian zeromean white-noise process W (t ) is filtered through two filters, whose
dynamic parameters are affected by the kind of soil. The two-sided
stationary PSD function of this model is
Su¨g (ω) =
ωg4
(ωg2 −
+ 4ζg2 ωg2 ω2
ω2)2 + 4ζg2 ωg2 ω2
ω4
Sw
(ωf2 − ω2)2 + 4ζ f2 ωf2 ω2
⎡ ˙ ˙ T⎤ =
Σ uu
˙ ˙ = E uu
⎢
⎥
⎣
⎦
2
0.141ζg u¨ g0
ωg 1 + 4ζg2
Σ u¨ A u¨ A = E ⎡u
¨ u
¨T⎤ =
⎢ A A⎥
⎣
⎦
(7)
Filter parameters can be calibrated so as to be representative of different soil conditions. In this paper, deterministic values of the filter
parameters are assumed as indicated in [141], which may be ideally
associated with stiff and soft soil profiles and, consequently, with excitations having different characteristics. The value of üg0 will be assumed equal to 0.3 g (g denoting the acceleration of gravity) throughout
the paper.
(11a)
˜ (ω) = RH
˜ U (ω) U¨g (ω) = HΔU (ω) U¨g (ω)
ΔU (ω) = RU
(11b)
ΔU˙ (ω) = iωΔU (ω) = iωHΔU (ω) U¨g (ω) = HΔU˙ (ω) U¨g (ω)
(11c)
A
(13)
u¨An = u¨n + u¨ g – this quantity is a measure of the vibration felt by the
occupants of the building
SPI2 = σu¨An;
(14)
• the standard deviation of the base shear V
T
b = τ Ku , which is related
to the cost of the foundations and to the entity of the seismic force
applied to the building [51]
SPI3 = σVb =
E [Vb2] =
E [τ T KuuT KT τ ] =
τ T K Σ uu KT τ ≡ k1 σu1;
(15)
• the average of the standard deviations of the interstory drift ratios
(IDRs) IDRj = Δuj / hj (with hj denoting the j th interstory height),
which is a measure of the global damage occurring in the infills [51]
SPI 4 =
1
n
n
∑ σIDRj =
j=1
1
n
n
∑
j=1
σΔuj
hj
;
drifts evaluated at the undamped natural frequency ω1 [25]
n
SPI5 =
∑ |HΔuj (ω1)|;
undamped fundamental frequency ω1 [54]
(11d)
SPI6 = |HVb (ω1)| = |τ T KHU (ω1)| ≡ k1 |Hu1 (ω1)|;
wherein the related transfer function vectors HU˙ , HΔU , HΔU˙ , HU¨A are
introduced.
Considering the transfer function vectors in Eqs. (10) and (11) and
the above stochastic definition of the earthquake ground motion as per
the PSD in (7), the random vibration theory produces the following set
of response covariance matrices (collecting the mean square response
quantities under a zero-mean stochastic seismic input)
˜ uu R
˜T
ΣΔUΔU = E[ΔUΔUT ] = RΣ
(18)
• the sum of the standard deviations of the damper forces f
dj = cdj f j Δu˙ j
as per Eq. (1), which is related to the overall cost of the added FVDs
[51]
nd
SPI fdtot =
∑ cdj f j σΔu˙ j.
j=1
∞
g
(17)
• the amplitude of the base shear transfer function evaluated at the
¨ A (ω) = U
¨ (ω) + τU¨g (ω) = (−ω2HU (ω) + τ ) U¨g (ω) = HU¨A (ω) U¨g (ω)
U
∫−∞ HU (ω) Su¨ (ω) HU (ω)*T dω
(16)
• the sum of the amplitudes of the transfer functions of the interstory
j=1
Σ uu = E[uuT ] =
(12e)
• the standard deviation of the top-story absolute floor acceleration
where HU (ω) is the n × 1 system displacement transfer function vector.
In a similar manner, it is possible to derive the system response in term
of velocity, interstory displacement, interstory velocity, and absolute
(total) floor acceleration respectively as
U˙ (ω) = iωHU (ω) U¨g (ω) = HU˙ (ω) U¨g (ω)
g
SPI1 = σun;
(9)
(10b)
A
standard deviation of the top-story displacement un – this
quantity is related to the overall displacement demand of the
structure and to the cost of the seismic joints [51]
with U(ω) and U¨g (ω) denoting the Fourier transform of u(t ) and u¨ g (t ) ,
respectively, i = −1 is the imaginary unit, and ω the natural circular
frequency. The system displacement response in the frequency domain
is defined by
HU (ω) = −[−ω2 M + i ω (C + Cd) + K]−1Mτ
∞
∫−∞ HU¨ (ω) Su¨ (ω) HU¨ (ω)*T dω
• the
3.2.1. Stochastic performance criteria
A set of quantities are introduced in this subsection that are useful
for subsequent comparisons among various design strategies proposed
in the literature. The equations of motion (6) can be rewritten in the
frequency domain as follows
(10a)
(12d)
are the expectation operator and the complex
in which E[·] and
conjugate transpose.
Based on the probabilistic framework adopted in this research work,
the performance indices are expressed as combination of elements of
the covariance matrices of the system response in (12) and, as such,
termed stochastic performance indices (SPIs) in line with other studies
from the literature [51,142,143]. The following set of stochastic performance indices are introduced, which may represent potential
quantities to minimize in the optimal design process of the FVDs (as will
be clarified in the following sections):
(8)
U (ω) = HU (ω) U¨g (ω)
(12c)
[·]*T
.
[−ω2 M + i ω (C + Cd) + K] U (ω) = −MτU¨g (ω)
g
˜ u˙ u˙ R
˜T
˙ ˙ T ] = RΣ
ΣΔUΔU
˙ ˙ = E[ΔUΔU
where ωg and ζg are the fundamental circular frequency and damping
ratio of the surface soil deposits, respectively, and ωf and ζ f represent
the Clough-Penzien filter parameters for the low-frequency adjustment
of the Kanai-Tajimi spectrum. The constant white-noise spectral level
Sw can be related to the bedrock peak ground acceleration (PGA) üg0
according to [140].
Sw =
∞
∫−∞ ω2HU (ω) Su¨ (ω) HU (ω)*T dω
(19)
(12a)
(12b)
As a general remark, SPI5 and SPI6 are related to amplitude of the
148
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
4. Design of fluid viscous dampers
convenience, FVDs are identical at every story. Despite its simplicity,
this method is likely to be not as effective as other more sophisticated
(non-uniform) distribution procedures for an equal total cost of the
dampers (in terms of cd tot ). Therefore, several procedures have been
developed in the literature to address the issue of damper placement
and sizing: some of the most popular ones are briefly outlined below.
4.1. Computation of the added damping
4.3. Distribution of the dampers along the building height
A practical concept to assess and quantify the effectiveness of the
damper distribution in a building is the evaluation of the modal
damping ratio associated with the supplemental FVDs. At a preliminary
stage of analysis and design, the viscous damping ratio of the added
FVDs in the first mode of vibration can be evaluated through the simplified formula proposed in the FEMA 356 [144] provisions, which is
here recalled and adapted to the matrix-vector notation of the present
paper:
In line with the sketch of Fig. 1, the design methodologies for the
distribution of the dampers along the building height can be grouped in
heuristic, analytical and evolutionary. Heuristic approaches exploit
some problem-specific knowledge of the structural behavior. Among
these methods, we mention the sequential placement, the story-shearproportional distribution and the stiffness-proportional distribution.
These methods are generally characterized by practicality and simplicity, thus they can be used without great computational effort and
without powerful software packages. Analytical approaches include the
methods based on optimal control theory, gradient-based search
methods, and a fully stressed analysis/redesign procedure. Finally, the
evolutionary methods, primarily genetic algorithms, based on natural
biological evolution, operate on a population of potential solutions
without computing any gradient. The fundamentals of these strategies
will be overviewed in the following subsections.
transfer function at the undamped fundamental frequency of the system
ω1. Therefore, unlike variance indicators, they are only indirectly related to the maximum system response, and this may not be the case for
narrow-band excitations whose frequency content is far from ω1.
n
ζd =
2
d
2
T
T1 ∑ j = 1 cdj f j (ϕ1j − ϕ1j − 1)
T ϕ RT DRϕ1
= 1 1 T
,
n
2
4π
4π ϕ1 Mϕ1
∑i = 1 mi ϕ1i
(20)
where ϕ1 is the first eigenvector of the undamped building structure
(such that Kϕ1 = Mϕ1 ω12 ), and T1 = 2π / ω1 is the fundamental period.
The simplified expression (20) was used in several research works
[145,61,76,121] and will be adopted later on in this paper. However, it
is worth noting that, strictly speaking, the actual modal damping ratios
induced by the supplemental dampers should be calculated from the
complex modal analysis of the nonclassically damped system —
whereas the eigenvectors entering Eq. (20) arise from the undamped
eigenvalue problem. Occhiuzzi [146] reviewed several design procedures proposed in the literature from the perspective of the corresponding achieved modal damping ratio, and showed that most of the
proposed design methods are associated with a first mode damping
ratio ζ d ≈ 20%. Higher values of the damping ratio, while implying a
modest reduction of the interstory drifts, may lead to a significant increase of the absolute accelerations. This is why in the sequel of the
paper a value of ζ d ≈ 20% is regarded as an optimum damping ratio to
achieve.
4.3.1. Heuristic approaches
The sequential search algorithm (SSA), first proposed by Zhang and
Soong [21], later applied by Wu et al. [24] to 3D torsionally-coupled
structures, independently validated by Shukla and Datta [33] in the
context of viscoelastic dampers, and subsequently simplified by Lopez
Garcia [37] and Soong [38] (SSSA), is perhaps one of the oldest and
pioneering approaches. More recent applications to 3D frame structures
was presented by Aguirre et al. [62]. The method was inspired by the
concept of a controllability index defined by Cheng and Pantelides
[150] in the field of active structural control. In each step of the SSA a
predefined amount of damping (through the addition of a supplemental
damper having fixed size) is added to the structure at the location in
which a specific controllability index (e.g., the interstory drift or the
interstory velocity) attains the largest value. The method implies the
use of dampers with the same size (i.e., having the same damping
coefficient cd ), which is convenient for practicality, since in realistic
implementations only a few damper sizes are to be adopted. Despite its
simplicity and the reasonable underlying physical meaning, this
method belongs to the class of repetitive methodologies (because the
dampers are sequentially added until a predefined cd tot is reached, or a
performance objective is met) and, thus, provides the optimal distribution of damping through an iterative process involving sets of timehistory analyses, which may be computationally demanding for structures with many DOFs. Furthermore, Singh and Moreschi [39] demonstrated that the SSA, primarily focused on controlling interstory
drifts, might be not really effective in reducing absolute floor accelerations, especially when considering buildings with non-uniform story
stiffness.
Some other distribution methods belonging to the heuristic class can
be listed here, which are based on a predefined added damping ratio ζ d
as per FEMA 356 [144], Eq. (20). These distribution methods, complying with the condition ζ d = ζ¯d , are:
4.2. Formulation of the design problem
Broadly speaking, the optimal damper design can be viewed as a
constrained optimization problem, which can be expressed in the following general mathematical form
min J (cdj ) subject to
cdj
g (cdj ) = g¯
(21)
where J is a specified objective function (or performance index, or cost
function) that is to be minimized and that depends upon the sought
damping coefficients cdj , and g is a given constrain function that, similarly, is related to the unknown damping coefficients and whose
value is to be equal to ḡ . From a mere mathematical viewpoint, the
problem stated in (21) represents a nonlinear constrained single-objective multivariable optimization problem, which can be solved
through a mathematical programming procedure, see e.g. [147–149]
for an up-to-date overview. For example, g could represent the sum of
n
the damping coefficients, i.e., g = ∑ j =d 1 cdj ≡ cd tot , so that the constraint
function in (21) represents a rough estimator of the dampers cost.
Further details on more appropriate functions of dampers cost proposed
in the literature are given in the following Section 4.4. Alternatively, if
the performance and effectiveness of the FVDs are deemed to be more
meaningful for the design process than the associated cost of the devices, the constraint function could represent a target damping ratio ζ d
resulting from the installation of the dampers, for instance computed
through the simplified Eq. (20).
The most trivial placement of FVDs is based on the “uniform distribution” (UD) assumption wherein, for design simplicity and
• the uniform distribution (UD) is the simplest one and assumes that the
damping coefficients cdj are identical at every storey, and equal to
n
cdj =
• the
149
∑i = 1 mi ϕ12i
4π ζ¯d
(j = 1, …, n);
nd
T1 ∑ j = 1 f j2 (ϕ1j − ϕ1j − 1)2
so-called
storey-shear-proportional-distribution
(22)
(SSPD)
was
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
arbitrary damping system, in an incremental manner [26]. On the basis
of a specific redesign procedure, Takewaki [25] developed an incremental method to determine the set of dynamic parameters (e.g. the
damping coefficients cdj of the added FVDs) corresponding to the
minimization of the sum of amplitudes of the interstory drifts transfer
functions evaluated at the undamped natural frequency ω1. Therefore,
the objective function considered by Takewaki is equal to SPI5 defined
in (17). A constraint condition on the sum of the damping coefficients
cd tot , representing the total cost of the FVDs, was also introduced in the
optimization problem. The optimization problem was solved by imposing the stationary conditions of the Lagrangian of the formulation, which gives rise to a sequential procedure for determining the
optimal distribution cdj of the problem, see also [56]. This incremental
procedure was later successfully applied to other mechanical problems
and sometimes termed “steepest direction search algorithm” [28] as the
sequential (updating) procedure was found to move towards the direction that more rapidly reduces the objective function under the (total
cost) constraint condition.
The redesign procedure and the incremental inverse problem approach were later used for the simultaneous identification of the optimal damping coefficients of the FVDs and of the story stiffness distribution of a shear building model [30] within a two-stage combined
displacement-acceleration control method. Additionally, the same numerical algorithm was also applied to planar moment-resisting frames
[29,36], to 3D shear building models [31], and to study optimal damper
placement considering soil-structure interaction [32]. The optimal
damper placement under uncertain ground acceleration was also investigated within a probabilistic framework through the concept of
critical excitation [35,67].
Aydin et al. used a variant of the Takewaki's optimization technique
by changing the objective function, focusing on the transfer function
amplitude of the base shear [54] and of the base moment [152]. Viola
and Guidi [136] extended the Takewaki's optimization technique by
also incorporating the supporting brace stiffness as a free variable to
optimize. These authors [136] proposed a two-step procedure: the first
step is aimed to identify the optimum damper distribution and the
optimum brace stiffness values by minimizing the sum of the meansquare interstory drifts, followed by a second optimization step to
minimize the maximum accelerations.
A fully-stressed analysis/redesign procedure was proposed by Levy and
Lavan [153]. This method uses an iterative procedure to maximize
(“fully stress”) the effect of the dampers on the performance index
parameter (e.g., interstory drift or acceleration response). The same
authors [154] later modified the original procedure by introducing a
constraint on the total damping. This procedure was extensively applied
to shear frames, industrial buildings and 3D irregular frames [50,70].
Quite recently Aguirre et al. [62] compared three different optimal
search procedures, including a min-max numerical optimization that
gives an exact solution, the SSSA [37,38], and the fully stressed analysis/redesign procedure [153]. The presented examples revealed that
the SSSA leads to a discrete approximation of the optimal solution (as
given by the min-max algorithm) that converges to the exact solution as
the step increments are progressively reduced. The comparison of the
different methodologies must also include practicality and computational cost considerations. The computational cost could be quantified
by the number of structural analyses involved to converge to the final
design. Whittle et al. [155] compared the effectiveness of five different
damper distributions: the uniform and stiffness proportional distributions, which are considered the simplest methods to implement (belonging to the heuristic class of methods), and three advanced methods,
namely the SSSA by Lopez Garcia [37], the inverse problem approach
by Takewaki [25] and the fully stressed analysis/redesign method by
Levy and Lavan [153]. These three advanced methods were selected
because they avoid the drawbacks of computationally-intensive
methods (for instance powerful optimization methods like genetic algorithms, described in the following subsection). Considering an
proposed by Pekcan et al. [34]. This distribution was motivated by
the observation of the reduced efficiency of the FVDs on the upper
stories due to the fact that the interstory velocities are typically
lower than those of the lower stories. It wan then proposed to distribute the FVDs in proportion to the design story shears Vsj . Considering that the value of Vsj at the storey j is proportional to the
n
parameter Sj = ∑i = j mi ϕ1i , the following distribution formula can be
derived [34,61]
n
cdj =
Sj ∑i = 1 mi ϕ12i
4π ζ¯d
(j = 1, …, n);
n
T1 ∑ j =d 1 Sj f j2 (ϕ1j − ϕ1j − 1)2
(23)
• the so-called storey-shear-strain-energy (SSSE) distribution was proposed by Hwang et al. [61], motivated by the concept of composite
damping ratio weighted by the element strain energy introduced by
Raggett [151]. Based on this concept, it was proposed to distribute
the FVDs in proportion to the storey shear strain energy, which can
be considered proportional to the parameter ψj = Sj (ϕ1j − ϕ1j − 1) ,
where Sj has already been introduced above. According to this criterion, the resulting distribution formula is
n
cdj =
ψj ∑i = 1 mi ϕ12i
4π ζ¯d
(j = 1, …, n);
n
T1 ∑ j =d 1 ψj f j2 (ϕ1j − ϕ1j − 1)2
(24)
• as a further refinement of the SSSE distribution procedure described
above, Hwang et al. [61] also proposed to distribute the FVDs only
to those stories whose shear strain energy ψj is larger than the
n
average story shear strain energy ψav = ∑i = 1 ψi / n . Let us denote with
neff the total number of these efficient stories, i.e., the stories such
that ψj > ψav . The storey-shear-strain-energy-to-efficient-storeys
(SSSEES) distribution formula is a variant of Eq. (24) applied to
these efficient storeys only, and with the sum on the denominator
restricted to these n eff efficient storeys only
n
cdj =
ψj ∑i = 1 mi ϕ12i
4π ζ¯d
(j = 1, …, n eff );
neff
T1 ∑ j = 1 ψj f j2 (ϕ1j − ϕ1j − 1)2
(25)
• within the class of Rayleigh damping systems [139,86], the stiffness-
proportional distribution (SPD) corresponds to a damping matrix
Cd = β K , which is represented by the “interstory installation
scheme” of the FVDs shown in Fig. 2 with a particular value of the
damping coefficients cdj . In order to guarantee the constraint ζ d = ζ¯d ,
it can easily be verified that the β constant must be equal to
β = 2ζ¯d/ ω1 [139,86].
4.3.2. Analytical approaches
Gluck et al. [22] adapted optimal control theory to the problem of
damper design aimed to minimize a performance cost function. The
optimal location of passive dampers was identified through different
approaches applicable to structures dominated by a single mode response. Applications to braced multistory buildings with viscous and
viscoelastic dampers were presented to prove the effectiveness of the
methodology.
Gradient-based search methods can be adopted provided the performance index is differentiable and its gradients can be computed.
Gradient-based
optimization
approaches
were
adopted
in
[18,25,39,43,48–50,125,70]. One of the most popular methods belonging to this class of methodologies is the inverse problem approach
developed by Takewaki. For a given set of dynamic parameters, including mass, stiffness and damping coefficients, it is easy to derive the
displacement transfer function HU (ω) through (10b). The inverse problem, i.e., deriving the set of dynamic parameters for a fixed (target)
transfer function is not as easy as the direct problem. The key idea of
Takewaki [25] was to solve such inverse problem, including an
150
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
is directly or indirectly related to a preassigned cost of the devices. The
latter may be either directly expressed through the cd tot value [25], or
indirectly considered through the use of a fixed damping ratio ζ d of the
added FVDs as per Eq. (20) [61]. Fixing ζ d prevents the cdj coefficients
from assuming disproportionately large values, which is very likely to
happen in an unconstrained optimization.
ensemble of 20 ground motions and both regular and irregular momentresisting frames, these authors concluded that all the five analyzed
distribution methods met the performance requirements in terms of
desired drift limits and reduced the floor accelerations; the three advanced methods achieved a comparable performance to each other,
slightly better than the two simplified approaches. Among the three
advanced methods, the fully stressed analysis/redesign method seemed
to be more convenient from a computational viewpoint, while the Takewaki damper distribution was achieved more quickly than the SSSA
method.
4.5. Performance objective
Based on the general design problem reported in Eq. (21), various
options were explored in the literature for the definition of the performance objective J, i.e., regarding the way the performance is evaluated. Among these choices, we mention:
4.3.3. Evolutionary approaches
Evolutionary methods, primarily genetic algorithms, automatically assume different starting points and perform the search sequentially
without computing any gradient. This could be an advantage for nondifferentiable performance index or for cost functions that vary steeply
over the range of the design variables so that several local optimum
values might exist. The underlying theory dates back to the early 70s,
and comprehensive treatments can be found in the books by Holland
[156], Goldberg [157] and Mitchell [158]. Based on principles of natural biological evolution, these methods basically operate on a population of potential solutions that tend towards the best individual
characterized by the best fitness value through random genetic modifications. These methods may suffer from an increased computational
effort as compared to gradient-based approaches.
Genetic algorithms were adopted in a number of research papers in
the context of damper design [40,45,47,44,159,51,58]. It is worth
pointing out the importance of the population size and the number of
evolutions allowed, as the probabilistic-based evolutionary methods
can converge toward the optimal configuration provided an appropriate
planning of the algorithm is made. Thus, the decision maker should
calibrate the above parameters based on problem-specific information
and knowledge, and extensive numerical simulations are often needed
to check the sensitivity of the solution [51].
• the maximum top floor displacement or the maximum interstory
shear [162] under white-noise seismic excitation;
• the time integral of the total mechanical energy (or Hamiltonian) of
•
•
•
•
•
•
•
•
4.4. Constraint condition
•
The usefulness of constraint conditions is related to the following
point. There are two competing aspects in the design, placement and
sizing of FVDs: one is the initial cost for the devices, and the other is the
reliability of the structural system, which is related to the seismic
performance of the controlled structure. In a simplified manner, the
initial cost of the damper system can be considered proportional to cd tot ,
which is frequently assumed in other papers from the literature, e.g.
[37,51,61,121]. Nevertheless, we point out that this is only a very
rough estimator of the actual dampers cost. More appropriate functions
of cost were developed in the literature. One of these was proposed by
Pollini et al. [160] and subsequently refined by the same authors in
[92]. This formulation incorporates three different contributions to the
cost function, namely a component Jl related to the number of locations
in which the dampers are installed (the installation of dampers implies
a preparation of the structure and an architectural constraint), a term Jm
related to the manufacturing cost of the dampers (proportional to the
peak damper force) and a third contribution Jp reflecting the prototypetesting cost associated with the requirements of modern seismic codes
(at least one damper is tested for each size group). On the other hand,
since the reliability of the structure is related to the cost associated with
earthquake losses and repairing effort after potential seismic events, in
the literature some studies have been proposed to find a trade-off between the two aspects, thus addressing the so-called life-cycle cost including the total expected cost during the entire structure lifetime
(initial, maintenance, and failure cost) within a so-called integrated
seismic design procedure, see e.g. [44,64,161,118,120] for some emblematic examples. Alternatively and using a much simpler optimization strategy, the two aspects may be simultaneously accounted for by
minimizing a seismic performance index under some constraint, which
•
•
•
•
•
the system [20], i.e., the sum of the kinetic and potential energy
contributions;
the control gain matrix in the field of active control theory [22];
the sum of story stiffness of a shear building subject to a set of FVDs
complying with an equal total cost constraint [23];
the sum of the amplitudes of the transfer functions of the interstory
drifts evaluated at the undamped natural frequency of the structure
[25];
the transfer function amplitude of the base shear evaluated at the
undamped natural frequency of the structure [54];
the transfer function amplitude of the base moment evaluated at the
undamped fundamental frequency ω1 [152];
the sum of the damping coefficients of the added dampers under a
target added damping ratio and interstory drift ratio [163];
a combination of stochastic performance criteria based on crossvariances of a variety of response indicators, see e.g. [45,51];
an energy-based global damage index (GDI) [48] that is the
weighted average index previously proposed by Bracci et al. [164]
in the context of nonlinear (yielding) controlled structures;
a weighted integral of squared interstory drift ratios of all stories
[49] for an ensemble of ground motions through an iterative procedure;
the maximum reduction in a predefined response indicator [39],
such as the base shear, interstory drifs, the overturning moment or
the floor acceleration [40];
the damper cost meant as the sum of the characteristic values (at
95% percentile) of the peak forces of the FVDs [119] under a constraint related to the maximum interstory drift ratio complying with
the Eurocode 8 provisions [165];
a multiobjective combined interstory drift-acceleration optimization
[58], or a combined displacement-acceleration criterion based on a
two-stage procedure [30];
the normalized life-cycle cost [44], or the life-cycle cost computed
as the sum of the initial cost (related to the size of the dampers) and
the expected life-cycle losses [161,118], or again a bi-objective
function including the competing mean total life-cycle cost and repair cost threshold [120];
a generalized performance index simultaneously incorporating displacements, absolute accelerations and base shear [166].
The particular selection of one (or more than one) performance
objective is strictly connected to the goal of the optimization procedure
and to some design choices that, in most cases, are problem-specific. As
an example, Liu et al. [47] proposed an iterative approach with heuristic improvement considering three different performance indices:
maximum interstory drift, absolute acceleration and device cost for
equal level of drift. The obtained damper distributions resulting from
the three design criteria were strongly dependent on the selected performance index. Aydin et al. [167] used the artificial bee colony
151
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 3. Brace-damper arrangements among the different bays investigated in various literature studies: a) Apostolakis and Dargush [174]; b) Mezzi [175]; c) Whittle
et al. [173].
opinion, these methods have some unique features in providing the
decision maker with a more complete set of solutions, thus allowing
trade-offs between different objectives. In this sense, they can be considered as superior and more general formulations as compared to
single-objective optimization approaches.
Furthermore, it is worth noting that the optimization problem set up
in [48,49] for both linear and nonlinear primary structures was formulated in a slightly different format than (21), as it involved multiple
inequality and equality constraints: the objective function was represented by the total cost of the FVDs ( J = cd tot ) and both the equations of motions and the performance index were reported as constraint
equality and inequality conditions, respectively. Similar concepts were
also applied to 3D structures in [50,93,70].
Considering the definition of performance indices under stochastic
excitation, although most procedures refer to statistical moment such as
standard deviation of the physical quantity of interest, reliability is
another critical aspect that should be considered, which was also noted
in [124]. As will be shown below, energy-based performance indices
imply the attainment of a trade-off between various indicators of the
structural response (interstory drifts, floor accelerations, etc). Under
this perspective, energy-based design philosophies can be somehow
optimization algorithm for the optimal placement of steel diagonal
braces, by considering four different objective functions, namely
transfer function amplitude of top displacement, top absolute acceleration, base shear and base moment. This method belongs to the wider
class of swarm intelligence based algorithms.
While almost all of the numerical procedures from the literature in
the field of damper placement and sizing were restricted to a singleobjective performance index, sporadic exceptions exist. In this regard,
Lavan and Dargush [58] adopted the concept of Pareto front [168] to
determine the optimal set of dampers (not only confined to FVDs, but
also buckling restrained devices and viscoelastic dampers) that minimized two performance indices. The two selected performance indices
were represented by the maximum inter-story drift and maximum absolute floor acceleration. In this way, the designer is able to control the
overall performance of the primary structure, non-structural components and contents [58]. Evolutionary methods underlying genetic algorithms were employed to evaluate all the Pareto front, i.e., all the
family of potential design solutions for which an improvement in one
objective function implies degradation in the other one. Similar multiobjective design was pursued very recently by Gidaris et al. [120] in a
more general life-cycle performance-based framework. In the authors’
152
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
viewed as optimization criteria of multiple objectives in a certain sense.
Previous investigations on the probabilistic criteria with respect to
multiple objectives and reliability can be reviewed from [169,170].
Ei (t ) =
∫0
t
− u̇ T Mτu¨ g dt; (input energy from the earthquake).
(27e)
The energy balance equation for unit time (or equation of power balance) is basically the same as Eq. (26), and is written as
4.6. Distribution of dampers among the different bays
e k (t ) + eds + edd (t ) + ee (t ) = ei (t )
Although in the present paper, and also in the majority of the literature studies, reference has been made to the search for an optimal
height-wise distribution of the dampers, in more realistic moment-resisting frames (MRFs) also the distribution of FVDs among different
bays may be of crucial importance. Indeed, while the damper maximum
forces are out-of-phase with displacements, there is an interim phase in
between peak displacement and peak velocity where the damper forces
sum up with the gravity loads. Thus, the increase of axial loads induced
by FVDs may cause overstressing of columns, especially for those supporting structural members located at lower stories and for higher
amounts of damping. As an example, Uriz and Whittaker [171] proved
that the seismic retrofit of a low-rise pre-Northridge steel MRF building
with added FVDs having around 40% damping led to a significant increase of column axial forces and base shear, which would require
supplemental strengthening of foundations and columns. This overloading of columns resulting from the addition of dampers is of key
importance in view of capacity design principles. A recent study by
Karavasilis [172] demonstrated that steel MRFs with FVDs are more
susceptible to column plastic hinging than analogous steel MRFs
without dampers, thus jeopardizing the occurrence of a global plastic
mechanism of the frame. In order to distribute the damper forces more
effectively avoiding overstressing of lower-story columns and detrimental effects on the plastic collapse mechanism of MRFs, diagonal
braces and multiple brace-damper arrangements can be adopted to
impose a desirable load path, as pointed out e.g. by Whittle et al. [173].
Among these studies on optimal brace-damper configurations per floor,
Apostolakis and Dargush [174] identified some alternated exterior/interior arrangements as effective configurations, Fig. 3a); Mezzi [175]
investigated 7 different configurations of energy-dissipating braces per
floor for an 18-story reinforced-concrete frame and found that the SP
brace layout minimized the axial forces among the non-random distributions, Fig. 3b); Whittle et al. [173] compared 5 different configurations, and demonstrated that combinations of braces placed to interior and exterior bays alternatingly are more effective than common
arrangements restricting braces to either interior or exterior bays,
Fig. 3c).
where the generic symbol e y denotes the rate of energy Ey at the time
instant t, i.e., the term within the integral of the corresponding energy
contribution Ey in (27). Since üg is treated as a stochastic process according to Section 3.2, the rate of total seismic input energy can be
stochastically evaluated by applying the expectation operator to the
terms in (28). In doing so, one should note that in stationary conditions
the expected value of the kinetic and elastic strain energy terms vanish,
from which we get
(28)
E[ei] = e¯i = E[eds] + E[edd]
(29a)
n
E[eds] = e¯ds = E[u˙ T Cu˙ ] =
n
∑ ∑ C(i,j) E[u˙ i u˙ j] = τT (C. *Σuu
˙ ˙ )τ
(29b)
i=1 j=1
n
E[edd] = e¯dd = E[u˙ T Cd u˙ ] =
n
∑ ∑ C(di,j) E[u˙ i u˙ j] = τT (Cd. *Σuu
˙ ˙ )τ
i=1 j=1
(29c)
where the symbol .* denotes the MATLAB element-wise multiplication
operator, and the symbol (·) represents the expectation operator applied
to (·) . The meaning of Eq. (29) is that, in stationary conditions, the
expected value of the input energy from the earthquake excitation is
split into two main contributions for a building structure with FVDs: the
energy dissipated by the structure Eds , which is related to the structural
damage of the building (since the overall dissipative mechanisms of the
structure are idealized through an equivalent viscous damping matrix
C ), and the energy dissipated by the FVDs Edd . The expected values of
the rates of such two energy terms, ēds and ēdd , are reported in (29b),
(29c). In an ideal scenario, if all the input energy were dissipated by the
FVDs, Ei = Edd , then ideally no damage would be expected in the main
structure as Eds = 0 . This could only occur for infinitely large values of
the damping coefficients cdj . However, due to the considerations made
in Section 4.4, some additional constraint conditions lead to more
realistic design scenarios.
5.1. Design of FVDs from energy-based perspectives
The basic observations outlined above leads to the following energybased design strategy for the FVDs that will be also explored in this
paper:
5. Energy-based perspectives for the assessment of the FVDs
performance
max e¯dd = τ T (Cd . *Σ uu
˙ ˙ )τ
Ek (t ) + Eds (t ) + Edd (t ) + Ee (t ) = Ei (t )
subject to
t
∫0
Eds (t ) =
∫0
Edd (t ) =
∫0
Ee (t ) =
∫0
t
u̇ T Mu
¨ dt =
t
t
1 T
u̇ (t ) Mu̇ (t ); (kinetic energy)
2
(26)
(27a)
u̇ T Cu̇ dt; (energy dissipated by the structure)
(27b)
u̇ T Cd u̇ dt; (energy dissipated by the FVDs)
(27c)
u̇ T Ku dt =
1 T
u (t ) Ku (t ); (elastic strain energy)
2
⎧ either ζ =
d
⎨
or
⎩
T T
T1 ϕ1 R DRϕ1
4π ϕ T Mϕ1
1
= ζ¯d
n
cdj = cd tot = ∑ j =d 1 =c¯d tot
(30b)
where ζ¯d and c̄d tot are a target performance index and a total cost index,
respectively, that are established a priori by the designer. Expressions
(30) represent a constrained optimization problem to solve. This energy-based approach was recently proposed by the authors in conjunction with a novel stochastic linearization technique for the design
of nonlinear FVDs [121]. It seems reasonable to include this method in
the comparative study performed in this paper, by considering the
simpler case of linear behavior of the viscous dampers.
The idea of maximizing the dissipated energy as a design approach
is obviously not new, and dates back to the late 80s with the landmark
work by Uang and Bertero [176]. Such an energy-based design approach is of general validity and applicable to supplemental dissipation
devices of any kind, not limited to FVDs, as reviewed for example by
Constantinou and Symans [177] and Soong and Dargush [2]. To quote
with the conventional meaning of the various energy contributions as
follows
Ek (t ) =
(30a)
cdj
In order to characterize the dynamic behavior of the schematic nstory building with FVDs shown in Fig. 2 from an energy-based perspective [176], the equations of motion (6) are multiplied by u̇T and
then integrated over the time, which yields the equation of relative
energy balance at the time instant t
(27d)
153
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Type B structure, have a uniform distribution of mass and a fundamental circular frequency ω1 = 5.39 rad/s . The Type A structure is
characterized by a uniform distribution of stiffness along the building
height, whereas the stiffness of the Type B structure is chosen such that
the transfer function amplitudes of the interstory drifts are uniform
[25]. All the interstory heights are set as hj = h = 3 m , in line with [51].
For these two structures, 3 different earthquake frequency contents
have been considered, associated with firm and soft conditions (corresponding to broad-band, and narrow-band excitations) and white-noise
random process as an idealization of an extremely broadband frequency
content. Furthermore, 2 different constraint conditions are considered
separately as presented in Eq. (30b), namely either a target damping
ratio ζ¯d = 20% as per FEMA 356 [144] Eq. (20), or a target total cost
index c¯d tot = nd ·1.5·106 = 9·106 Ns/m , according to [25]. Therefore, 12
different input data are considered. To assess the performance of each
system, the first 6 stochastic performance indices introduced in Section
3.2.1, Eqs. (13)–(18), are utilized in the following normalized format
˜ i = SPIi (i = 1, …, 6)
SPI
SPIi(0)
(31)
where SPIi(0)
indicates the value of the stochastic performance index SPIi
in the uncontrolled configuration, i.e., for the system without FVDs.
˜ i are expected to be lower than one, and the lower
Therefore, all the SPI
values correspond to the more effective strategies. Furthermore, in
order to normalize the energy-based response indicator ēdd , a so-called
“filtered energy index” is here introduced, which represents the portion
of the global seismic input energy that is not dissipated by the FVDs and
thus penetrates (or filters) into the building structure:
Fig. 4. Comparative performance histograms of design methodologies of FVDs
for Type B structure: a) firm soil conditions; b) soft soil conditions; c) whitenoise seismic input.
FEI =
just a few examples, energy-balance concepts and energy-based design
methodologies were applied to buckling restrained braces [178], to
hysteretic dampers [179], to maximize the performance of non-conventional tuned mass damper (TMD) implemented via inter-story isolation [180], or to improve the seismic performance of base-isolated
buildings with TMD at basement [181] and TMD coupled with inertial
dampers [182–185]. The list of papers reported above is certainly not
exhaustive.
In the context of the energy-based design criteria of FVDs earlier
proposed in the literature, it is worth mentioning the method proposed
by Sorace and Terenzi [186]. In the quoted study, the damping coefficients of nonlinear FVDs (with fractional power law of velocity) at a
particular j th story were selected to ensure a certain, predefined amount
of energy dissipation capacity. More specifically, this dissipation capacity was expressed in terms of a ratio between the energy dissipated
by the FVDs at a particular story EDj , and the absolute input energy at
that story EIj , namely βj = EDj / EIj . Although the energy dissipation capacity is inherently a function of time, for practicality the energy balance condition was enforced at the instant corresponding to the maximum interstory drift occurrence. This energy-based design criterion
was validated against full-scale experimental tests on reinforced concrete and steel structures. Furthermore, values of the target, postcalculated, and experimentally recorded βj energy ratios were in reasonably good agreement with one another, which confirmed that this
energy-based design procedure was effective for the attainment of a
predefined seismic performance level (like a performance drift index).
e¯i − e¯dd
e¯
= 1 − dd .
e¯i
e¯i
(32)
The proposed design strategy reported in (30) can also be viewed as
a constrained minimization of the FEI given in Eq. (32). To obtain
realistic, non-zero values of the FEI , the inherent damping of the frame
is assumed to generate a damping ratio ζ = 0.05 for all the vibration
modes (this is slightly in contrast to the original input data by Takewaki, who instead neglected the inherent damping of the frame). Finally, in order to keep an eye to the overall cost of the damper system,
not only the cd tot value, but also the SPI fdtot indicator as per Eq. (19)
have been computed for each design strategy.
In the following subsections we will present 13 different distribution
methods for FVDs under a target damping ratio constraint, and other 10
different distribution methods for FVDs under an equal total cost index
constraint. Therefore, multiplying the different distribution methods by
the sets of input data, an overall number of 138 different design scenarios have been considered in this comparative study.
6.1. Comparison of distribution methods based on target damping ratio
constraint
In Fig. 4 the values of the six SPIi and the FEI are reported through
comparative histograms for 6 different design strategies that were described above. Firm, soft soil conditions and white-noise seismic input
are considered individually for the Type B structure (results for the
Type A structure are comparable, and not reported for brevity). These
seven response indicators can reasonably be considered an overall
measure of the global seismic performance, since they include displacements, accelerations, forces and energy-specific quantities. By inspection of Fig. 4, the overall seismic performance of the 6 designs
strategies seems to be more or less comparable. This is reasonable since
the added damping ratio (to the first fundamental mode) in all cases is
identical and equal to ζ¯d = 20% for all the 6 strategies. However, scrutinizing the various response indicators more in-depth reveals some
little differences. For example, it can be noted that the design strategies
UD and SSSEES give slightly higher values of displacements and absolute accelerations. In all cases the design strategy FEI performs very
6. Comparison of different design methodologies under stochastic
excitation
The aim of this Section is to compare different methodologies for the
design of FVDs. The comparison is made with reference to two simple
benchmark six-story shear-type structures already discussed in the
Takewaki 1997 paper [25] and addressed in other research works, e.g.
[41,51]. These two structures, in the sequel denoted as Type A and
154
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 5. Product of all the SPIs for different design methodologies under equal ζ d = ζ¯d constraint.
added damping ratio ζ¯d = 20%. Nevertheless, some improvements are
noted in the energy-based design strategy minimizing the FEI, especially with regard to absolute floor acceleration profiles as compared to
the strategies SSPD, SSSE, SSSEES. It is worth noting that the design
strategy SSSEES leads to the placement of dampers to the first three
stories only for both Type A and B structures. This placement leads to
the same damping ratio ζ¯d = 20% as the other procedures, and indeed
the displacement profiles are comparable. However, the response in
terms of interstory drifts and absolute floor accelerations, especially for
the non-regular-in-elevation Type B structure, increases significantly
for the higher stories wherein no dampers are installed.
The corresponding distributions of the damping coefficients according to the different design methodologies are reported in Fig. 7. It
can be noted that: 1) the values of cdj decrease with increasing the story
level for SSPD design, which is reasonable since Sj is maximum at the
first floor and minimum at the top floor for a uniform mass distribution;
2) the values of cdj are mostly concentrated at the first three floors for
SSSE design, since in such three stories the maximum story shear strain
energy occurs for the given structural parameters and input data; 3) for
similar reasons, the values of cdj according to the SSSEES design are
uniquely concentrated in the first three stories, as already noted above;
4) as to the SPD design, considering the stiffness matrix of the Type A
and B structures, the values of cdj are uniformly distributed in the
former case and they decrease with increasing the story level in the
latter case; 6) the cdj values of the (energy-based) FEI design are nearly
uniform in the case of Type A structure, and exhibit variations along the
building height for the Type B structure.
It is also worth noting that, while the simplified design procedures
UD, SSPD, SSSE, SSSEES and SPD are independent on the earthquake
frequency content and give the same damping coefficient distribution
regardless of the seismic input, the energy-based procedure incorporates the influence of the earthquake frequency content (through
well and leads to the lowest values of almost all the considered response
indicators. This means that, for a given target damping ratio, the distribution of the damping coefficients according to the energy-based
performance criterion stated in (30) is more effective than other distribution methods since it achieves the maximum dissipation of the
seismic input energy, thus implying the minimization of the filtered
energy index as per Eq. (32). This assessment is clearly highlighted in
Fig. 5 wherein the product of all the considered performance indices
7
˜ i (here the FEI is assumed as the seventh normalized SPI, i.e.,
∏i = 1 SPI
˜ 7 ) is illustrated for the 6 design strategies. From this figure, it
FEI ≡ SPI
can be clearly seen that, for a variety of structural parameters and
seismic input data, the design strategy minimizing the FEI represents
the best distribution method among the set of design methodologies
complying with a target damping ratio constraint. Indeed, this strategy
7
˜ i . This
is associated with the lowest value of the product index ∏i = 1 SPI
conclusion corroborates the importance of energy-based perspectives in
optimizing the performance of passive dissipation systems in general, as
already documented in several textbooks in the relevant literature
[2,9].
The results may be compared in slightly different formats in terms of
profiles of some response variables along the building height. Indeed,
associated with the damping coefficients cdj identified by the different
design methodologies, the response covariance matrices can be computed through Eqs. (12). This leads to the determination of the profiles
of standard deviations of displacements, interstory drifts and absolute
floor accelerations illustrated in Fig. 6 for firm soil conditions. The
results for other soil conditions, here not reported for the sake of
brevity, follow similar trends and leads to qualitatively similar conclusions. As stated above, there is a reasonable agreement of the seismic
response of the different methodologies, no matter how the damping
coefficients are distributed. This is due to the common value of the
155
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 6. Profiles of standard deviations of displacements, interstory drifts, and absolute floor accelerations for different design methodologies in firm soil conditions.
brevity). The UD design produces the highest cost indices, which may
be due to the fact that in this very simple distribution the FVDs are not
efficiently “engaged” to dissipate the seismic energy from the earthquake. This means that FVDs of higher size are needed to achieve the
same seismic performance as the other procedures if these devices are
distributed uniformly. The FEI design gives damping coefficients that
are lower or reasonably in line with the other procedures. This confirms
that the energy-based procedure is effective not only in terms of performance, but also from economical perspectives. The lowest value of
the total cost indices is associated with the SSSEES design. This result is
consistent with the conclusions of Hwang et al. [61]. However, for the
its PSD function) on the damping coefficients. The influence of different
seismic input characteristics on the cdj values is documented in Fig. 8.
As an example, if soft soil conditions are considered, no FVDs should be
placed in the top story of Type A structure, which is rather different
from both firm soil conditions and white-noise seismic input.
It is also interesting to check whether the different methodologies
imply a different total cost of the damper system to achieve a comparable seismic performance. To this aim, in Fig. 9 both the SPI fdtot
indicator (as per Eq. (19)) and the cd tot value are reported for each
design strategy for the Type A structure and soft soil conditions (similar
results are obtained for other configurations and are not reported for
156
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 7. Damper coefficient distribution for different design methodologies in firm soil conditions.
Fig. 8. Influence of the earthquake frequency content on the damper coefficient distribution according to the energy-based design (minimization of the FEI).
157
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 9. Comparison of cost indices of the damper system for different design methodologies in soft soil conditions and Type A structure.
best trade-off between performance and cost is given by the design
˜ 6 and the FEI — the other design strategies are
strategies minimizing SPI
less convenient either because of lower performance or due to the
higher associated cost; 2) for the Type B structure the cost associated
with almost all the design strategies is more or less comparable (except
˜ 6 ), but the best performance is offered
˜ 3 and SPI
the ones minimizing SPI
˜ 5 and the FEI . These results pertain to
by the strategies minimizing SPI
soft soil conditions, but qualitatively similar conclusions may be drawn
for other earthquake frequency contents that are not reported here for
the sake of brevity. These outcomes further confirm the previous
statement: among a set of possible performance indices to minimize,
selecting the damping coefficients of the FVDs from energy-based perspectives (i.e., by minimizing the FEI ) seems to be a convenient choice
since it provides an effective trade-off between overall performance and
cost of the damper system.
selected input data, the seismic performance of the SSSEES design
seems to be a bit poorer than the other procedures, cf. again Figs. 5 and
6.
Based on the above results, it is concluded that the design of FVDs
from energy-based perspectives is an effective strategy to maximize the
dissipation behavior of the devices and to minimize the system response
accordingly. It could be argued whether the energy-based indicator FEI
is the best performance index to minimize in the optimization problem
(21). An appropriate answer to this question could only be given by
selecting alternative performance indices different from the FEI in the
optimization problem (21), and comparing the resulting seismic performance. In other words, the maximum performance reduction that
can be obtained given the assumed constraint is evaluated for performance indices other than the energy dissipated by the dampers. In this
way, it is possible to quantify what is the maximum response reduction
achievable and how effective are the different design procedures. Reference is still made to the SPIi reported in Section 3.2.1. In this way, in
addition to the previous 6 strategies discussed above, 7 further design
strategies are here investigated, which minimize the SPIi (i = 1, …, 6)
that are defined in (13) – (18), and the SPI fdtot defined in (19). These
strategies thus represent alternative design methodologies proposed in
the literature by different authors as stated in Section 3.2.1. The purpose of this comparison is to verify which performance index leads to
the best overall vibration control of the structure among a set of 8
different possible criteria, including the FEI . It is also important to
understand the trade-off between performance and cost of each
strategy. To this aim, for each of these 8 different design strategies we
have additionally computed a normalized cost indicator defined as:
)
c˜d(ktot
=
cd(k)tot
cd(max)
tot
6.2. Some remarks on distribution methods based on total cost constraint
As an alternative to the previous strategies based on a target
damping ratio constraint, a total cost constraint in the form of a given
cd tot value could be assumed in the optimization problem (30). Without
going into the detailed comparisons discussed in the previous subsection, the main result of this further investigation is the following: for
c̄d tot = 9·106 Ns/m , which is the same value as in the examples discussed by Takewaki [25], the numerical optimization problem (30)
with total cost constraint for white-noise input excitation provides the
same set of damping coefficients identified by Takewaki [25]. In other
words, the damping coefficients found by Takewaki [25] through the
inverse problem approach are also the ones that, subject to the c̄d tot
constraint, maximize the energy dissipated by the FVDs ēdd , thereby
minimizing the FEI. It can be seen from Fig. 12 that such design
strategy gives a quite balanced and effective reduction of displacement¯ 4 ) response. Based on
¯ 1), acceleration- (SPI
¯ 2 ), and IDR-related (SPI
(SPI
the previous remarks, it can be easily demonstrated that the Takewaki
design strategy, which gives the lowest value of the FEI, also corresponds to the lowest product of the stochastic performance indices
7
˜ i.
∏i = 1 SPI
(k = 1, …, 8)
(33)
cd(max)
tot
where
is the maximum total cost index among the 8 design
max ) = 1, and c̃ (k ) < 1 for all the
strategies for k = k max , for which c̃d(ktot
d tot
other 7 remaining design strategies with k ≠ k max . Relevant results are
summarized in Fig. 10 in an histogram format for both Type A and B
structures.
The maximum c̃d tot value (equal to one) occurs for the strategy
˜ 1 and SPI
˜ 3 for Type A and Type B structure, respectively.
minimizing SPI
All the other design strategies yield c̃d tot values lower than one, as expected. It can be noted that each design strategy has its own minimum
˜ i value corresponding to the specific response indicator that is adSPI
dressed in the optimization problem. However, an overall control
considering the whole set of response indicators, i.e., not just limited to
a single response indicator, would be highly desirable. In this regard,
the desired design strategy should give a balanced reduction of a group
˜ i . As already stated above, this overall control performance could
of SPI
be well represented by the product of all the performance indices
7
˜ i . In order to assess the trade-off between the overall perfor∏i = 1 SPI
mance and the cost of each strategy, all the design strategies are displayed by a characteristic point in a combined cost-versus-performance
plane in Fig. 11. It can be noticed that: 1) for the Type A structure the
7. Seismic performance evaluation via time-history analysis
The comparative study among the different design methodologies is
then further developed via time-history analysis [187]. In particular,
the FEMA P695 far-field record set [188] comprising 44 historically
recorded seismic input components (relevant to 22 pairs of ground
motions) is considered. These earthquake inputs are selected from the
PEER NGA database using some record selection criteria as specified in
the Appendix A.7 in [188]. The PGA of these earthquake ground motions ranges from 0.21 g to 0.82 g , with an average PGA of 0.43 g . The
optimal FVDs are identified by the constrained optimization problem
(30) with a target damping ratio constraint equal to ζ¯d = 20%. Considering the characteristics of the individual ground motion samples,
158
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 10. Comparison of design methodologies based on different performance indices in soft soil.
ζ¯d = 20%, the distribution of FVDs according to the energy-based procedure leads to reductions of around 50% of displacements and IDRs, up
to 70 − 80% of absolute floor accelerations and floor shear forces as
compared to the uncontrolled scenario without FVDs. In order to
highlight and summarize the performance of the energy-based design
procedure against other simplified design methodologies based on the
same damping ratio ζ¯d = 20%, in Table 1 some scalar quantities are
listed. In particular, the mean values over the building height of dis¯ , absolute accelerations ü¯A , shear forces f¯s and
placements ū , IDRs IDR
the value of the base shear Vb are computed not only for the energybased design strategy (minimizing the FEI in (32)) and the uncontrolled
case (without FVDs), but also for the UD, SSPD, SSSE and SSSEES distributions introduced in Section 6.1. All these strategies (except for the
uncontrolled case) are characterized by the same damping ratio. It can
be seen that the energy-based procedure minimizing the FEI leads to a
very effective reduction of response indicators including displacements,
IDRs, accelerations and forces as compared to the uncontrolled case.
For the given target damping ratio constraint, it seems that this procedure yields the best vibration control, at least among the procedures
the firm-soil Kanai-Tajimi PSD function has been considered for the
optimization problem, in line with other literature studies that considered the FEMA P695 far-field record set [189]. Indeed, all the 44
recorded samples are pertinent to site class C (soft rock/very dense soil)
or D (stiff soil) conditions according to the NEHRP classification. Both
the individual response spectra and the median response spectrum (for
a 0.05 damping ratio) of this ensemble of seismic inputs are reported in
Fig. 13. A variety of response indicators have been computed for each
ground motion input. Average (over the 44 earthquake ground motions) maximum (MAX) values of a few of the most important response
indicators are reported in this paper.
In particular, the seismic performance of the Type B structure
without dampers and with FVDs optimized from energy-based perspectives is compared in Fig. 14. The averaged profiles (considering the
44 samples) of the MAX displacements (normalized by the last-floor
displacement in the uncontrolled case without FVDs u6(0) ), MAX interstory drift ratios IDRs, MAX floor absolute accelerations and MAX
floor shear forces (normalized by the total structural weight Wtot ) are
depicted. It can be noted that, for a given damping ratio constraint
Fig. 11. Assessment of the trade-off between performance and cost of each strategy based on different performance indices in soft soil.
159
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Fig. 12. Performance bar plots of different design methodologies based on total cost constraint for Type B structure and white-noise seismic input.
attenuating the peak values that occur in the uncontrolled structure in
the first instants of the seismic event comprised in the range [5−10] s.
Similar considerations may be drawn for other ground motion samples
here not shown for brevity.
Finally, some observations related to the energy balance are worth
discussing, since this is pertinent to the energy-based perspectives of the
present paper. To this aim, in Fig. 16 the different energy contributions
entering the equation of energy balance (26)–(27) are plotted as a
function of time, in order to capture the evolution of the different energy terms in two different scenarios. The upper part of the Fig. 16
refers to the design of the FVDs from energy-based perspectives,
thereby minimizing the FEI in (32), whereas the lower part concerns the
SSSEES distribution according to Eq. (25). We note that the seismic
input energy Ei is slightly different in the two cases, because it depends
on the relative velocity of the system u̇ through the integral in (27e).
Regardless of the specific value of the seismic input energy, it is worth
analyzing how this energy is distributed in the various portions according to the energy balance equation (26). In this regard, it is noted
that the kinetic and elastic strain energy contributions, Ek and Ee , are
confined to just a few seconds of the ground motion time-history
Fig. 13. Pseudo-acceleration response spectra for the 44 historically recorded
ground motions of the FEMA P695 far-field record set [188].
analyzed in this study.
An example of time-history response, in terms of the last-floor displacement us6 and base shear Vb , is shown in Fig. 15 for an arbitrary
accelerogram (specifically, for the last sample of the FEMA P695 farfield record set, namely record ID 121712, corresponding to the Tolmezzo 1976 ground motion occurred in Friuli, Italy). It can be noticed
that the FVDs are effective in dampening the response and in
Fig. 14. Seismic performance of the Type B structure with energy-based distribution of FVDs under the FEMA P695 far-field record set [188] (average MAX values on
44 samples).
160
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
Table 1
Comparative assessment of different damper distributions: average MAX values on 44 samples of the FEMA P695 far-field record set [188].
response quantity
design strategy for equal damping ratio
(average values on 44 samples)
UD
SSSE
SSPD
SSSEES
FEI
no FVDs
5.86
0.55
3.06
5.46
0.54
2.99
5.53
0.54
2.99
5.52
0.67
3.61
5.41
0.54
2.99
mean floor shear force f¯s [kN]
204.21
172.95
179.67
269.79
168.74
489.23
base shear Vb [kN]
938.99
840.84
862.95
741.05
823.02
1509.90
mean displacement ū [cm]
¯ [%]
mean interstory drift ratio IDR
mean absolute acceleration ü¯A [m/s2]
10.01
1.22
7.23
Fig. 15. Time-history response in terms of last-floor displacement and base shear under the Tolmezzo 1976 earthquake ground motion (record ID 121712 [188]).
definitely confirms that the energy dissipation performance of the FVDs
may be superior even with dampers of lower size provided the devices
are distributed according to more effective schemes that are optimized
from an energy-based perspective.
(nearly in the aforementioned range [5−10] s), and then they are almost negligible after t = 10 s as the intense part of the record is concluded, cf. Fig. 15. What remain are the energy contributions dissipated
by the structure Eds and by the FVDs Edd , whose sum nearly equals Ei as
the mechanical energy (Ek + Ee ) vanishes. In the energy-based design,
most of the seismic input energy is dissipated by the FVDs, thereby the
remaining part (dissipated by the structure) is limited, whereas in the
SSSEES distribution the value of Edd is not as high as in the previous
case, which implies that the complementary value Eds (representing the
damage in the structure) must increase in order to balance the seismic
input energy. This fact can be quantified by the value of the FEI as per
(32), which is equal to 0.18286 in the former case and to 0.3616 in the
latter case. In other words, in the former case only the 18% of the total
seismic input energy has to be dissipated by the structure as the remaining 82% has been effectively absorbed by the FVDs. On the contrary, implementing FVDs that are not optimized from energy-based
perspectives, like in the SSSEES distribution case, may yield a poor
dissipation behavior, meaning that the structure must dissipate a larger
amount of energy from the earthquake excitation. As a result, it is expected that distributions of FVDs that do not dissipate a great amount of
energy yield higher response quantities than other distributions procedures optimized from energy-based perspectives. Interestingly, the
value of cd tot is 1.58·107 Ns/m in the former case (energy-based design)
and 1.72·107 Ns/m in the latter case (SSSEES distribution). This
8. Concluding remarks
The design of FVDs for earthquake protection of building structures
has been the main subject of this review paper. Several design philosophies exist in the literature, which are based upon different design
criteria, performance indices, constraint conditions, and numerical algorithms. The fundamentals of the most popular design strategies
proposed in the literature have been reviewed in the first part of this
paper through the aid of an illustrative sketch summarizing the organization of this review, in order to provide some preliminary background through an overview of methodologies. A wide group of analytical, heuristic, and numerical methodologies have been outlined in a
very concise manner, along with their underlying theoretical principles,
the involved performance criteria and the related computational algorithms. Key aspects and main characteristics of each strategy have been
outlined. Although the primary aim of FVDs is to dissipate the largest
possible amount of energy from the earthquake excitation, most of the
design strategies proposed in the literature are not explicitly based on
energy concepts, but rather they address some response quantities that
161
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
reasonably low cost of the damper system in comparison with other
strategies. Similar conclusions have been drawn in the time-history
analyses under the FEMA P695 far-field record set. Corresponding to a
20% damping ratio constraint condition of the energy-based design
strategy, reductions of around 50% of displacements and IDRs, up to
70 − 80% of the absolute floor accelerations and floor shear forces have
been obtained as compared to the uncontrolled scenario without FVDs.
Despite the large number of design scenarios considered, several
simplifications have been adopted for the comparative study, which
should be acknowledged here to draw possible directions of future
work. The first assumption regards the structural model employed in
the numerical examples — cf. again the sketch in Fig. 1 and the possible
options for the designer. Planar shear-type frames with linear elastic
behavior, equipped with linear FVDs mounted on infinitely stiff braces
represent the simplest option for the analysis among the possible alternatives. More realistic 3D structures (having more DOFs than planar
frames and, possibly, coupled torsional behavior), undergoing elastoplastic deformations under strong earthquakes, in conjunction with a
more sophisticated model for the damper-brace system certainly require further investigation. Another important aspect concerns the
evaluation of the dampers cost, which in this paper has been roughly
associated to two simplified measures like the sum of the damping
coefficients cd tot and the sum of the damper forces (in stochastic terms)
SPI fdtot . In this regard, more appropriate cost functions accounting not
only for the damper forces, but also for the installation costs, prototypetesting costs, or life-cycle costs including initial, maintenance and
failure costs would be desirable. Re-examination of the analyzed design
strategies under these more realistic modeling assumptions represents a
challenging line of future research.
Fig. 16. Energy contributions under the Tolmezzo 1976 earthquake ground
motion (record ID 121712 [188]) with FVDs: a) energy-based design; b) SSSEES
distribution.
Acknowledgements
are only indirectly consequent of the actual energy dissipated by these
devices.
Motivated by the above idea, this paper has focused on specific
energy-based perspectives for the design of FVDs. The energy balance
equations have been resorted to and rephrased in stochastic terms, by
considering the uncertain nature of the earthquake excitation through
the Kanai-Tajimi PSD function. The seismic performance of the reviewed design methodologies can thus be re-interpreted from energybased perspectives, by assessing the amount of energy dissipated by the
FVDs, out of the total input energy from the earthquake excitation.
Building on these concepts, a numerical strategy has been used to
identify the sets of FVDs that attain the best energy dissipation behavior
among the possible sets of dampers complying with a desired (target)
damping ratio, or corresponding to a fixed total cost of the damper
system.
A broad comparative study among different design methodologies
has been carried out. This comparative study has comprised: 1) stochastic dynamic analysis considering a stochastic seismic input; 2) timehistory analysis under an ensemble of 44 earthquake ground motions
selected from the FEMA P695 far-field record set. With regard to the
stochastic dynamic analysis, an extensive comparative study has been
performed, which has comprised 2 structures, 3 earthquake frequency
contents, 2 constraint conditions, and 23 distributions of FVDs, for an
overall number of 138 different design scenarios. It has been found that
the energy-based design strategy provides the best distribution method
among a wide group of analyzed design methodologies complying with
an equal (target) damping ratio constraint. Indeed, the energy-based
strategy has led to a global control of the seismic response including
displacements, accelerations, forces and energy-specific quantities. It
has been demonstrated that the newly introduced “filtered energy
index” is the best performance index to minimize in the optimization
problem out of a set of 8 possible choices. Additionally, by analyzing
the trade-off between performance and cost, it has been found that the
energy-based design strategy is effective not only in terms of performance, but also from economical perspectives since it involves
The authors have particularly appreciated the several constructive
comments made by one of the 5 anonymous reviewers on an earlier
version of this manuscript.
References
[1] Housner GW, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF,
Skelton RE, Soong TT, Spencer BF, Yao JTP. Structural control: past, present, and
future. J Eng Mech 1997;123(9):897–971.
[2] Soong TT, Dargush GF. Passive energy dissipation systems in structural engineering. Chichester: John Wiley & Sons; 1997.
[3] Soong TT, Spencer BF. Supplemental energy dissipation: state-of-the-art and stateof-the-practice. Eng Struct 2002;24:243–59.
[4] Naeim F, Kelly JM. Design of seismic isolated structures: from theory to practice.
New York: John Wiley & Sons; 1999.
[5] De Domenico D, Ricciardi G, Benzoni G. Analytical and finite element investigation
on the thermo-mechanical coupled response of friction isolators under bidirectional excitation. Soil Dyn Earth Eng 2018;106:131–47.
[6] De Domenico D, Ricciardi G. Improving the dynamic performance of base-isolated
structures via tuned mass damper and inerter devices: a comparative study. Struct
Control Health Monit 2018;25(10):e2234.
[7] Hart GC, Wong K. Structural dynamics for structural engineers. New York: John
Wiley and Sons; 2000.
[8] Hanson RD, Soong TT. Seismic design with supplemental energy dissipation devices. Oakland, CA: EERI; 2001.
[9] Christopoulos C, Filiatrault A. Principles of passive supplemental damping and
seismic isolation. Pavia, Italy: IUSS Press; 2006.
[10] Lee D, Taylor DP. Viscous damper development and future trends. Struct Des Tall
Build 2001;10(5):311–20.
[11] Pekcan G, Mander JB, Chen SS. The seismic response of a 1:3 scale model RC
structure with elastomeric spring dampers. Earthq Spectra 1995;11(2):249–67.
[12] Taylor DP. Damper retrofit of the London Millennium Footbridge a case study in
biodynamic design. Taylor Devices, Inc.; 2002.
[13] Sorace S, Terenzi G. Non-linear dynamic modelling and design procedure of FV
spring-dampers for base isolation. Eng Struct 2001;23(12):1556–67.
[14] Sorace S, Terenzi G. Fluid viscous damper-based seismic retrofit strategies of steel
structures: general concepts and design applications. Adv Steel Constr
2009;5(3):325–42.
[15] Pekcan G, Mander JB, Chen SS. Fundamental considerations for the design of nonlinear viscous dampers. Earthq Eng Struct Dyn 1999;28:1405–25.
[16] Terenzi G. Effetti dissipativi nell'isolamento sismico [Damping Effects in the
Seismic Isolation] [Ph.D. dissertation]. Florence, Italy: University of Florence;
162
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
1994. (in Italian).
[17] Lin WH, Chopra AK. Earthquake response of elastic SDF systems with non-linear
fluid viscous dampers. Earthq Eng Struct Dyn 2002;31(9):1623–42.
[18] De Silva CW. An algorithm for the optimal design of passive vibration controllers
for flexible systems. J Sound Vib 1981;75(4):495–502.
[19] Ashour SA, Hanson RD. Elastic response of buildings with supplemental damping.
Report UMCE 1987:871.
[20] Gürgöze M, Müller PC. Optimal positioning of dampers in multi-body systems. J
Sound Vib 1992;158(3):517–30.
[21] Zhang RH, Soong TT. Seismic design of viscoelastic dampers for structural applications. J Struct Eng 1992;118(5):1375–92.
[22] Gluck N, Reinhorn AM, Gluck J, Levy R. Design of supplemental dampers for
control of structures. J Struct Eng 1996;122(12):1394–9.
[23] Tsuji M, Nakamura T. Optimum viscous dampers for stiffness design of shear
buildings. Struct Des Tall Build 1996;5(3):217–34.
[24] Wu B, Ou JP, Soong TT. Optimal placement of energy dissipation devices for threedimensional structures. Eng Struct 1997;19(2):113–25.
[25] Takewaki I. Optimal damper placement for minimum transfer functions. Earthq
Eng Struct Dyn 1997;26(1113):1124.
[26] Takewaki I. Efficient redesign of damped structural systems for target transfer
functions. Comput Methods Appl Mech Eng 1997;147(3–4):275–86.
[27] Fu Y, Kasai K. Comparative study of frames using viscoelastic and viscous dampers.
J Struct Eng 1998;124(5):513–22.
[28] Takewaki I. Optimal damper positioning in beams for minimum dynamic compliance. Comput Methods Appl Mech Eng 1998;156(14):363–73.
[29] Takewaki I, Yoshitomi S. Effects of support stiffnesses on optimal damper placement for a planar building frame. Struct Des Tall Build 1998;7(4):323–36.
[30] Takewaki I. Displacement-acceleration control via stiffness-damping collaboration.
Earthq Eng Struct Dyn 1999;28(12):1567–85.
[31] Takewaki I, Yoshitomi S, Uetani K, Tsuji M. Non-monotonic optimal damper placement via steepest direction search. Earthq Eng Struct Dyn 1999;28(6):655–70.
[32] Takewaki I, Uetani K. Optimal damper placement for building structures including
surface ground amplification. Soil Dyn Earth Eng 1999;18(5):363–71.
[33] Shukla AK, Datta TK. Optimal use of viscoelastic dampers in building frames for
seismic force. J Struct Eng 1999;125(4):401–9.
[34] Pekcan G, Mander JB, Chen SS. Design and retrofit methodology for building
structures with supplemental energy dissipating systems. Report No. MCEER-990021. New York: Multidisciplinary Center for Earthquake Engineering Research,
State University of New York at Buffalo, 1999.
[35] Takewaki I. Optimal damper placement for critical excitation. Prob Eng Mech
2000;15:317–25.
[36] Takewaki I. Optimal damper placement for planar building frames using transfer
functions. Struct Multidiscip Optim 2000;20(4):280–7.
[37] Lopez Garcia D. A simple method for the design of optimal damper configurations
in MDOF structures. Earthq Spectra 2001;17(3):387–98.
[38] Lopez Garcia D, Soong TT. Efficiency of a simple approach to damper allocation in
MDOF structures. Struct Contr Health Monit 2002;9(1):19–30.
[39] Singh MP, Moreschi LM. Optimal seismic response control with dampers. Earthq
Eng Struct Dyn 2001;30(4):553–72.
[40] Singh MP, Moreschi LM. Optimal placement of dampers for passive response
control. Earthq Eng Struct Dyn 2002;31:955–76.
[41] Trombetti T, Silvestri S. Added viscous dampers in shear-type structures: the effectiveness of mass proportional damping. J Earthq Eng 2004;8(2):275–313.
[42] Trombetti T, Silvestri S. On the modal damping ratios of shear-type structures
equipped with Rayleigh damping systems. J Sound Vib 2006;292(1–2):21–58.
[43] Park JH, Kim J, Min KW. Optimal design of added viscoelastic dampers and supporting braces. Earthq Eng Struct Dyn 2004;33(4):465–84.
[44] Park KS, Koh HM, Hahm D. Integrated optimum design of viscoelastically damped
structural systems. Eng Struct 2004;26(5):581–91.
[45] Wongprasert N, Symans MD. Application of a genetic algorithm for optimal
damper distribution within the nonlinear seismic benchmark building. J Eng Mech
2004;130(4):401–6.
[46] Lee KS, Fan CP, Sause R, Ricles J. Simplified design procedure for frame buildings
with viscoelastic or elastomeric structural dampers. Earth Eng Struct Dyn
2005;34(10):1271–84.
[47] Liu W, Tong M, Lee GC. Optimization methodology for damper configuration
based on building performance indices. J Struct Eng 2005;131(11):1746–56.
[48] Lavan O, Levy R. Optimal design of supplemental viscous dampers for irregular
shear-frames in the presence of yielding. Earthq Eng Struct Dyn
2005;34(8):889–907.
[49] Lavan O, Levy R. Optimal design of supplemental viscous dampers for linear
framed structures. Earthq Eng Struct Dyn 2006;35(3):337–56.
[50] Lavan O, Levy R. Optimal peripheral drift control of 3D irregular framed structures
using supplemental viscous dampers. J Earth Eng 2006;10(6):903–23.
[51] Silvestri S, Trombetti T. Physical and numerical approaches for the optimal insertion of seismic viscous dampers in shear-type structures. J Earthq Eng
2007;11(5):787–828.
[52] Trombetti T, Silvestri S. Novel schemes for inserting seismic dampers in shear-type
systems based upon the mass proportional component of the Rayleigh damping
matrix. J Sound Vib 2007;302(3):486–526.
[53] Bhaskararao AV, Jangid RS. Optimum viscous damper for connecting adjacent
SDOF structures for harmonic and stationary white-noise random excitations.
Earthq Eng Struct Dyn 2007;36(4):563–71.
[54] Aydin E, Boduroglu MH, Guney D. Optimal damper distribution for seismic rehabilitation of planar building structures. Eng Struct 2007;29(2):176–85.
[55] Lee SH, Park JH, Lee SK. Allocation and slip load of friction dampers for a
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[65]
[66]
[67]
[68]
[69]
[70]
[71]
[72]
[73]
[74]
[75]
[76]
[77]
[78]
[79]
[80]
[81]
[82]
[83]
[84]
[85]
[86]
[87]
[88]
163
seismically excited building structure based on story shear force distribution. Eng
Struct 2008;30(4):930–40.
Takewaki I. Building control with passive dampers: optimal performance-based
design for earthquakes. Singapore: John Wiley & Sons; 2009.
Lagaros ND, Plevris V, Mitropoulou CC. Design optimization of active and passive
structural control systems. USA: IGI Global; 2013.
Lavan O, Dargush GF. Multi-objective evolutionary seismic design with passive
energy dissipation systems. J Earth Eng 2009;13(6):758–90.
Weng DG, Zhang C, Lu XL, Zeng S, Zhang SM. A simplified design procedure for
seismic retrofit of earthquake-damaged RC frames with viscous dampers. Struct
Eng Mech 2012;44(5):611–31.
Zhou Y, Lu X, Weng D, Zhang R. A practical design method for reinforced concrete
structures with viscous dampers. Eng Struct 2012;39:187–98.
Hwang JS, Lin WC, Wu NJ. Comparison of distribution methods for viscous
damping coefficients to buildings. Struct Infrastruct Eng 2013;9(1):28–41.
Aguirre JJ, Almazán JL, Paul CJ. Optimal control of linear and nonlinear asymmetric structures by means of passive energy dampers. Earthq Eng Struct Dyn
2013;42(3):377–95.
Martínez CA, Curadelli O, Compagnoni ME. Optimal design of passive viscous
damping systems for buildings under seismic excitation. J Constr Steel Res
2013;90:253–64.
Castaldo P, De Iuliis M. Optimal integrated seismic design of structural and viscoelastic bracing-damper systems. Earthq Eng Struct Dyn 2014;43(12):1809–27.
Seo CY, Karavasilis TL, Ricles JM, Sause R. Seismic performance and probabilistic
collapse resistance assessment of steel moment resisting frames with fluid viscous
dampers. Earth Eng Struct Dyn 2014;43(14):2135–54.
Alibrandi U, Falsone G. Optimal design of dampers in seismic excited structures by
the expected value of the stochastic dissipated power. Prob Eng Mech
2015;41:129–38.
Fujita K, Moustafa A, Takewaki I. Optimal placement of viscoelastic dampers and
supporting members under variable critical excitations. Earth Struct
2010;1(1):43–67.
Murakami Y, Noshi K, Fujita K, Tsuji M, Takewaki I. Simultaneous optimal damper
placement using oil, hysteretic and inertial mass dampers. Earth Struct
2013;5(3):261–76.
Kojima K, Fujita K, Takewaki I. Review: building earthquake resilience in sustainable cities in terms of input energy. Sust Cities Soc 2014;12:46–62. https://doi.
org/10.1016/j.scs.2014.01.004.
Lavan O. Optimal design of viscous dampers and their supporting members for the
seismic retrofitting of 3D irregular frame structures. J Struct Eng
2015;141(11):04015026.
Mori C, Sorace S, Terenzi G. Seismic assessment and retrofit of two heritage-listed
R/C elevated water storage tanks. Soil Dyn Earth Eng 2015;77:123–36.
Kasagi M, Fujita K, Tsuji M, Takewaki I. Automatic generation of smart earthquake-resistant building system: hybrid system of base-isolation and buildingconnection. J Heliyon 2016;2(2). https://doi.org/10.1016/j.heliyon.2016.e00069.
[Article e00069].
Taniguchi M, Fujita K, Tsuji M, Takewaki I. Hybrid control system for greater
resilience using multiple isolation and building connection. Front Built Environ
(Earth Eng) 2016:2. [Article 26].
Hao LF, Zhang RF. Structural safety redundancy-based design method for structure
with viscous dampers. Struct Eng Mech 2016;59(5):821–40.
Kawamoto T, Fujita K, Tsuji M, Takewaki I. Robust optimal placement of dampers
in structures with set-back and eccentricity using sensitivity analysis for integrated
transfer function. Int J Earth Impact Eng 2016;1(4):377–94.
Kandemir-Mazanoglu EC, Mazanoglu K. An optimization study for viscous dampers between adjacent buildings. Mech Syst Sign Proc 2017;89:88–96.
Sorace S, Terenzi G. Existing prefab R/C industrial buildings: seismic assessment
and supplemental damping-based retrofit. Soil Dyn Earth Eng 2017;94:193–203.
Tamura G, Taniguchi M, Fujita K, Tsuji M, Takewaki I. Optimal damper placement
in hybrid control system of multiple isolation and building connection. Int J Earth
Impact Eng 2017;2(1):67–87.
Tani T, Maseki R, Takewaki I. Innovative seismic response controlled system with
shear wall and concentrated dampers in lower stories. Front Built Environ (Earthq
Eng) 2017;3:57. [Article].
Altieri D, Tubaldi E, Patelli E, Dall Asta A. Assessment of optimal design methods
of viscous dampers. Procedia Eng 2017;199:1152–7.
Impollonia N, Palmeri A. Seismic performance of buildings retrofitted with nonlinear viscous dampers and adjacent reaction towers. Earth Eng Struct Dyn
2018;47(5):1329–51.
Wang S, Mahin SA. High-performance computer-aided optimization of viscous
dampers for improving the seismic performance of a tall building. Soil Dyn Earth
Eng 2018;113:454–61.
Taylor DP, Tonawanda NY. Toggle linkage seismic isolation structure, Patent no.
5934028; 1999.
Zhang R, He H, Weng D, Zhou H, Ding S. Theoretical analysis and experimental
research on toggle-brace-damper system considering different installation modes.
Sci Iran 2012;19(6):1379–90.
Şigaher AN, Constantinou MC. Scissor-jack-damper energy dissipation system.
Earthq Spectra 2003;19(1):133–58.
Chopra AK. Dynamics of structures. 4th edn USA: Pearson Prentice Hall; 2012.
Tubaldi E, Barbato M, Ghazizadeh S. A probabilistic performance-based risk assessment approach for seismic pounding with efficient application to linear systems. Struct Saf 2012;36:14–22.
Tubaldi E, Barbato M, Dall'Asta A. Performance-based seismic risk assessment for
buildings equipped with linear and nonlinear viscous dampers. Eng Struct
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
[120] Gidaris I, Taflanidis AA, Mavroeidis GP. Multiobjective design of supplemental
seismic protective devices utilizing lifecycle performance criteria. J Struct Eng
2018;144(3):04017225.
[121] De Domenico D, Ricciardi G. Earthquake protection of structures with nonlinear
viscous dampers optimized through an energy-based stochastic approach. Eng
Struct 2019;179:523–39.
[122] Di Paola M, La Mendola L, Navarra G. Stochastic seismic analysis of structures
with nonlinear viscous dampers. J Struct Eng 2007;133(10):1475–8.
[123] De Domenico D, Ricciardi G. Improved stochastic linearization technique for
structures with nonlinear viscous dampers. Soil Dyn Earth Eng 2018;113:415–9.
[124] Zeng X, Peng Y, Chen J. Serviceability-based damping optimization of randomly
wind-excited high-rise buildings. Struct Des Tall Spec Build 2017;26(11):e1371.
[125] Attard TL. Controlling all interstory displacements in highly nonlinear steel
buildings using optimal viscous damping. J Struct Eng 2007;133(9):1331–40.
[126] Lavan O, Cimellaro GP, Reinhorn AM. Noniterative optimization procedure for
seismic weakening and damping of inelastic structures. J Struct Eng
2008;134(10):1638–48.
[127] Cimellaro GP, Lavan O, Reinhorn AM. Design of passive systems for control of
inelastic structures. Earth Eng Struct Dyn 2009;38(6):783–804.
[128] Singh MP, Verma NP, Moreschi LM. Seismic analysis and design with Maxwell
dampers. J Eng Mech 2003;129:273–82.
[129] Chen YT, Chai YH. Effects of brace stiffness on performance of structures with
supplemental Maxwell model-based brace-damper systems. Earth Eng Struct Dyn
2011;40(1):75–92.
[130] Castellano MG, Colato GP, Infanti S. Use of viscous dampers and shock transmission units in the seismic protection of buildings. In: Proceedings of the 13th
world conference on earthquake engineering. Vancouver, Canada; 2004.
[131] Zhou Y, Li H. Analysis of a high-rise steel structure with viscous damped outriggers. Struct Des Tall Spec Build 2014;23(13):963–79.
[132] Seleemah A, Constantinou MC. Investigation of seismic response of buildings with
linear and nonlinear fluid viscous dampers. Report No. NCEER 97 0004, Buffalo,
N.Y.; 1997.
[133] Akcelyan S, Lignos DG, Hikino T, Nakashima M. Evaluation of simplified and stateof-the-art analysis procedures for steel frame buildings equipped with supplemental damping devices based on E-defense full-scale shake table tests. J Struct
Eng 2016;142(6):1–17.
[134] Park JH, Kim J, Min KW. Optimal design of added viscoelastic dampers and
supporting braces. Earth Eng Struct Dyn 2004;33(4):465–84.
[135] Ou JP, Long X, Li QS. Seismic response analysis of structures with velocity-dependent dampers. J Const Steel Res 2007;63(5):628–38.
[136] Viola E, Guidi F. Influence of the supporting braces on the dynamic control of
buildings with added viscous dampers. Struct Contr Health Monit
2009;16(3):267–86.
[137] Uang CM, Bertero VV. Evaluation of seismic energy in structures. Earth Eng Struct
Dyn 1990;19:77–90.
[138] Chen JB, Liu W, Peng Y, Li J. Stochastic seismic response and reliability analysis of
base-isolated structures. J Earthq Eng 2007;11:903–24.
[139] Clough RW, Penzien J. Dynamics of structures. 3rd edition Berkeley, CA, USA:
Computers and Structures Inc.; 2003.
[140] Buchholdt H. Structural dynamics for engineering. London: Thomas Teldfort;
1997.
[141] Der Kiureghian A, Neuenhofer A. Response spectrum method for multi-support
seismic excitations. Earthq Eng Struct Dyn 1992;21:713–40.
[142] Zhang R, Zhao Z, Pan C. Influence of mechanical layout of inerter systems on
seismic mitigation of storage tanks. Soil Dyn Earth Eng 2018;114:639–49.
[143] Pan C, Zhang R. Design of structure with inerter system based on stochastic response mitigation ratio. Struct Contr Health Monit 2018;25(6):e2169.
[144] Federal Emergency Management Agency (FEMA) and American Society of Civil
Engineers (ASCE). Prestandard and commentary for the seismic rehabilitation of
buildings. Report FEMA-356, Washington, DC, November 2000.
[145] Ramirez OM, Constantinou MC, Kircher CA, Whittaker AS, Johnson MW, Gomez
JD, Chrysostomou CZ. Development and evaluation of simplified procedures for
the analysis and design of buildings with passive energy dissipation systems.
Technical Report MCEER-00-0010 Revision 1, November 16, 2001, Univ at
Buffalo, NY, 2001.
[146] Occhiuzzi A. Additional viscous dampers for civil structures: analysis of design
methods based on effective evaluation of modal damping ratios. Eng Struct
2009;31(5):1093–101.
[147] Haftka RT, Gürdal Z. Elements of structural optimization. Dordrecht: Kluwer
Academic; 1992.
[148] Rao SS. Engineering optimization. New York, NY: Wiley; 1996.
[149] Boyd S, Vandenberghe L. Convex optimization. Cambridge UK: Cambridge
University Press; 2004.
[150] Cheng FY, Pantelides CP. Optimal placement of actuators for structural control.
Technical report NCEER-88-0037, National Center for Earthquake Engineering
Research, State Univ. of New York at Buffalo, Buffalo, N.Y, 1988.
[151] Raggett JD. Estimating damping of real structures. J Struct Div
1975;101(9):1823–35.
[152] Aydin E. Optimal damper placement based on base moment in steel building
frames. J Constr Steel Res 2012;79:216–25.
[153] Levy R, Lavan O. Fully stressed design of passive controllers in framed structures
for seismic loadings. Struct Multidiscip Optim 2006;32:485–98.
[154] Lavan O, Levy R. Simple iterative use of Lyapunov's solution for the linear optimal
seismic design of passive devices in framed buildings. J Earth Eng
2009;13(5):650–66.
[155] Whittle JK, Williams MS, Karavasilis TL, Blakeborough A. A comparison of viscous
2014;126:90–9.
[89] Tubaldi E. Dynamic behavior of adjacent buildings connected by linear viscous/
viscoelastic dampers. Struct Control Health Monit 2015;22(8):1086–102.
[90] Licari M, Sorace S, Terenzi G. Nonlinear modeling and mitigation of seismic
pounding between R/C frame buildings. J Earth Eng 2015;19(3):431–60.
[91] Pratesi F, Sorace S, Terenzi G. Analysis and mitigation of seismic pounding of a
slender R/C bell tower. Eng Struct 2014;71:23–34.
[92] Pollini N, Lavan O, Amir O. Minimum-cost optimization of nonlinear fluid viscous
dampers and their supporting members for seismic retrofitting. Earthq Eng Struct
Dyn 2017;46(12):1941–61.
[93] Lavan O, Amir O. Simultaneous topology and sizing optimization of viscous
dampers in seismic retrofitting of 3D irregular frame structures. Earth Eng Struct
Dyn 2014;43(9):1325–42.
[94] Sorace S, Terenzi G. The damped cable system for seismic protection of frame
structures Part I: general concepts, testing and modeling. Earth Eng Struct Dyn
2012;41(5):915–28.
[95] Sorace S, Terenzi G. The damped cable system for seismic protection of frame
structures Part II: design and application. Earth Eng Struct Dyn
2012;41(5):929–47.
[96] Naeem A, Kim J. Seismic retrofit of a framed structure using damped cable systems. Steel Comp Struct 2018;29(3):287–99.
[97] Pekcan G, Mander JB, Chen SS. Experiments on steel MRF building with supplemental tendon system. J Struct Eng 2000;126(4):437–44.
[98] Pekcan G, Mander JB, Chen SS. Balancing lateral loads using tendon-based supplemental damping system. J Struct Eng 2000;126(8):896–905.
[99] Adachi F, Yoshitomi S, Tsuji M, Takewaki I. Nonlinear optimal oil damper design
in seismically controlled multi-story building frame. Soil Dyn Earth Eng
2013;44:1–13.
[100] Lang ZQ, Guo PF, Takewaki I. Output frequency response function based design of
additional nonlinear viscous dampers for vibration control of multi-degree-offreedom systems. J Sound Vib 2013;332(19):4461–81.
[101] Bahnasy A, Lavan O., Linear or nonlinear fluid viscous dampers? A seismic point of
view. In: Proceedings of the structures congress, Pittsburgh; 2013.
[102] Fujita K, Kasagi M, Lang ZQ, Guo PF, Takewaki I. Optimal placement and design of
nonlinear dampers for building structures in the frequency domain. Earthq Struct
2014;7(6):1025–44.
[103] Rama Raju K, Ansu M, Iyer NR. A methodology of design for seismic performance
enhancement of buildings using viscous fluid dampers. Struct Contr Health Monit
2014;21(3):342–55.
[104] Dall'Asta A, Tubaldi E, Ragni L. Influence of the nonlinear behavior of viscous
dampers on the seismic demand hazard of building frames. Earth Eng Struct Dyn
2016;45(1):149–69.
[105] Parcianello E, Chisari C, Amadio C. Optimal design of nonlinear viscous dampers
for frame structures. Soil Dyn Earth Eng 2017;100:257–60.
[106] Jamshidiha HR, Yakhchalian M, Mohebi B. Advanced scalar intensity measures for
collapse capacity prediction of steel moment resisting frames with fluid viscous
dampers. Soil Dyn Earth Eng 2018;109:102–18.
[107] Akcelyan S, Lignos DG, Hikino T. Adaptive numerical method algorithms for
nonlinear viscous and bilinear oil damper models subjected to dynamic loading.
Soil Dyn Earth Eng 2018;113:488–502.
[108] Logotheti VE, Kafetzi TC, Papagiannopoulos GA, Karabalis DL. On the use of interstorey velocity for the seismic retrofit of steel frames with viscous dampers. Soil
Dyn Earth Eng 2018. https://doi.org/10.1016/j.soildyn.2018.07.042.
[109] Lin YY, Tsai MH, Hwang JS, Chang KC. Direct displacement-based design for
building with passive energy dissipation systems. Eng Struct 2003;25(1):25–37.
[110] Sullivan TJ, Lago A. Towards a simplified direct DBD procedure for the seismic
design of moment resisting frames with viscous dampers. Eng Struct
2012;35:140–8.
[111] Diotallevi PP, Landi L, Dellavalle A. A methodology for the direct assessment of
the damping ratio of structures equipped with nonlinear viscous dampers. J Earth
Eng 2012;16(3):350–73.
[112] Landi L, Fabbri O, Diotallevi PP. A two-step direct method for estimating the
seismic response of nonlinear structures equipped with nonlinear viscous dampers.
Earth Eng Struct Dyn 2014;43(11):1641–59.
[113] Martinez-Rodrigo M, Romero ML. An optimum retrofit strategy for moment resisting frames with nonlinear viscous dampers for seismic applications. Eng Struct
2003;25(7):913–25.
[114] Silvestri S, Gasparini G, Trombetti T. A five-step procedure for the dimensioning of
viscous dampers to be inserted in building structures. J Earthq Eng
2010;14(3):417–47.
[115] Palermo M, Muscio S, Silvestri S, Landi L, Trombetti T. On the dimensioning of
viscous dampers for the mitigation of the earthquake-induced effects in momentresisting frame structures. Bull Earth Eng 2013;11(6):2429–46.
[116] Silvestri S, Palermo M, Trombetti T. A direct procedure for the seismic design of
frame structures with added viscous dampers. Int J Saf Secur Eng
2017;7(4):498–509.
[117] Palermo M, Silvestri S, Landi L, Gasparini G, Trombetti T. A “direct five-step
procedure” for the preliminary seismic design of buildings with added viscous
dampers. Eng Struct 2018;173:933–50.
[118] Gidaris I, Taflanidis AA. Performance assessment and optimization of fluid viscous
dampers through life-cycle cost criteria and comparison to alternative design approaches. Bull Earth Eng 2015;13(4):1003–28.
[119] Navarra G, Lo Iacono F, Oliva M. Probabilistic optimal design of passive control
devices coherently with seismic codes response spectra. In: Proceedings of
AIMETA 2017, The Italian Association of Theoretical and Applied Mechanics,
Salerno, Italy 4–7 September 2017.
164
Soil Dynamics and Earthquake Engineering 118 (2019) 144–165
D. De Domenico et al.
[156]
[157]
[158]
[159]
[160]
[161]
[162]
[163]
[164]
[165]
[166]
[167]
[168]
[169]
[170]
[171]
[172]
[173]
[174]
frames with hysteretic passive devices. Earth Eng Struct Dyn 2010;39(4):355–76.
[175] Mezzi M. Innovative configurations and morphologies using dissipating bracing
systems. In: Proceedings of 9th U.S. national and 10th Canadian conference on
earthquake engineering, Toronto, Canada, 2010.
[176] Uang C, Bertero V. Use of energy as a design criterion in earthquake-resistant
design. Technical Report UCB/EERC 88-18, Earthquake Engineering Research
Center, University of California at Berkeley, 1988.
[177] Constantinou MC, Symans MD. Seismic response of structures with supplemental
damping. Struct Des Tall Build 1993;2(2):77–92.
[178] Kim J, Choi H, Chung L. Energy-based seismic design of structures with bucklingrestrained braces. Steel Comp Struct 2004;4(6):437–52.
[179] Benavent-Climent A. An energy-based method for seismic retrofit of existing
frames using hysteretic dampers. Soil Dyn Earth Eng 2011;31(10):1385–96.
[180] Reggio A, De Angelis M. Optimal energy-based seismic design of non-conventional
Tuned Mass Damper (TMD) implemented via inter-story isolation. Earthq Eng
Struct Dyn 2015;44(10):1623–42.
[181] De Domenico D, Ricciardi G. Earthquake-resilient design of base isolated buildings
with TMD at basement: application to a case study. Soil Dyn Earth Eng
2018;113:503–21.
[182] Pietrosanti D, De Angelis M, Basili M. Optimal design and performance evaluation
of systems with Tuned Mass Damper Inerter (TMDI). Earthq Eng Struct Dyn
2017;46(8):1367–88.
[183] De Domenico D, Ricciardi G. An enhanced base isolation system equipped with
optimal tuned mass damper inerter (TMDI). Earthq Eng Struct Dyn
2018;47(5):1169–92.
[184] De Domenico D, Impollonia N, Ricciardi G. Soil-dependent optimum design of a
new passive vibration control system combining seismic base isolation with tuned
inerter damper. Soil Dyn Earth Eng 2018;105:37–53.
[185] De Domenico D, Ricciardi G. Optimal design and seismic performance of tuned
mass damper inerter (TMDI) for structures with nonlinear base isolation systems.
Earthq Eng Struct Dyn 2018;47(12):2539–60.
[186] Sorace S, Terenzi G. Seismic protection of frame structures by fluid viscous
damped braces. J Struct Eng 2008;134(1):45–55.
[187] De Domenico D, Falsone G, Ricciardi G. Improved response-spectrum analysis of
base-isolated buildings: a substructure-based response spectrum method. Eng
Struct 2018;162:198–212.
[188] FEMA P695 (Federal Emergency Management Agency). Quantification of building
seismic performance factors. Federal Emergency Management Agency,
Washington, D.C.; 2009.
[189] Anajafi H, Medina RA. Comparison of the seismic performance of a partial mass
isolation technique with conventional TMD and base-isolation systems under
broad-band and narrow-band excitations. Eng Struct 2018;158:110–23.
damper placement methods for improving seismic building design. J Earth Eng
2012;16(4):540–60.
Holland JH. Adaptation in natural and artificial systems. an introductory analysis
with applications to biology, control and artificial intelligence. Ann Arbor, USA:
The University of Michigan Press; 1975.
Goldberg DE. Genetic algorithms in search; optimization and machine learning.
MA: Addison-Wesley, Reading; 1989.
Mitchell M. An introduction to genetic algorithms. Cambridge, MA: MIT Press;
1996.
Movaffaghi H, Friberg O. Optimal placement of dampers in structure using genetic
algorithm. Eng Comput 2006;23(6):597–606.
Pollini N, Lavan O, Amir O. Towards realistic minimum-cost optimization of viscous fluid dampers for seismic retrofitting. Bull Earth Eng 2016;14(3):971–98.
Shin H, Singh M. Minimum failure cost-based energy dissipation system designs
for buildings in three seismic regions Part II: application to viscous dampers. Eng
Struct 2014;74:275–82.
Constantinou MC, Tadjbakhsh IG. Optimum design of a first story damping system.
Comput Struct 1983;17(2):305–10.
Aydin E. A simple damper optimization algorithm for both target added damping
ratio and interstorey drift ratio. Earth Struct 2013;88:083–109.
Bracci JM, Reinhorn AM, Mander JB, Kunnath SK. Deterministic model for seismic
damage evaluation of RC structures. Technical Report NCEER-89-0033, 1989.
European Committee for standardization. Eurocode 8: Design of structures for
earthquake resistance - Part 1 (Final draft prEN 1998-1).Brussels; 2003.
Cimellaro GP. Simultaneous stiffness-damping optimization of structures with
respect to acceleration, displacement and base shear. Eng Struct
2007;29(11):2853–70.
Aydin E, Sonmez M, Karabork T. Optimal placement of elastic steel diagonal
braces using artificial bee colony algorithm. Steel Comp Struct
2015;19(2):349–68.
Pareto V. Manuel d′économie politique. Paris: Giard; 1927.
Peng YB, Ghanem R, Li J. Generalized optimal control policy for stochastic optimal
control of structures. Struct Contr Health Monit 2013;20(2):187–209.
Li J, Peng YB, Chen JB. Probabilistic criteria of structural stochastic optimal
controls. Prob Eng Mech 2011;26(2):240–53.
Uriz P, Whittaker AS. Retrofit of pre-Northridge steel moment-resisting frames
using fluid viscous dampers. Struct Des Tall Build 2001;10(5):371–90.
Karavasilis TL. Assessment of capacity design of columns in steel moment resisting
frames with viscous dampers. Soil Dyn Earth Eng 2016;88:215–22.
Whittle JK, Williams MS, BlakeBorough A. Performance of structural members in
seismic retrofitted frames with viscous dampers. In: Proceedings of 15WCEE,
Lisboa 2012.
Apostolakis G, Dargush GF. Optimal seismic design of moment-resisting steel
165