# Fuzzy Structural Analysis of a Tuned Mass Damper Subject to Random Vibration

Authors: Giuseppe Carlo Marano; Emiliano Morrone; Giuseppe Quaranta; Francesco Trentadue
Journal: Advances in Acoustics and Vibration
 (2008)

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## Abstract

The uncertainty is a typical feature of each
human activity since the greatest part of the information is always affected by a sure
level of scattering. Different methodologies which deal with the uncertainty of the real
problems exist. The principal aim of this paper is to present
an innovative hybrid approach which combines fuzzy and stochastic theories in
facing the structural analysis of a tuned mass damper subject to a dynamic random
load, modelled by a modulated filtered white noise. In this work the parameters
involved in the structural analysis will be considered uncertain and supposed fuzzy
sets to take into account the effects of lexical and informal uncertainties which
cannot be studied in a probabilistic way. The system analysis is conducted by
means ofα-level optimization technique.
Successively, a numerical example is presented to show the effectiveness of the
proposed procedure. Moreover, a sensitivity analysis is performed to expose the
variation of the structural response membership function considering different
input values. Finally, a comparison between the response nominal value and the
fuzzificated one is proposed to obtain a structural amplification factor.


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## Body

## 1. Introduction

The limitation of the effects of the vibrations due to the environmental
dynamic loads is a very important matter in structural engineering. In this
field, many different strategies have been proposed, especially with regard to
the problems of the random vibrations induced by natural or artificial loads,
as those due to the earthquakes, wind pressure, traffic vibrations, sea waves,
and so on. Generally, in literature, four groups of control systems are
distinguished: active, hybrid, semiactive, and passive. Among these, the passive
systems are the most unsophisticated and cheapest. These systems are
characterized by the fact that they completely manage and dissipate energy in a
mechanical way [1]. For this reason, they have several important
advantages: the systems are purely mechanical, often simplistic, and do not
require external power or device that is an important source of robustness. The
passive systems have been used successfully in many real cases. Among numerous
passive control available devices, the tuned mass damper (TMD) is frequently
adopted both in mechanical systems and in civil structures, in new
constructions as well as in the retrofit of existing buildings. In general, the
purpose of installing a TMD is to guarantee a suitable level of protection in
the primary structure to assure an adequate safety level, both for the
structure for its contents regarding a defined limit state. Moreover, TMD is introduced to reduce
the discomfort to occupants and/or to limit the damage of the equipments in
particular into high-rise buildings, especially when moderate (and frequent)
seismic loads are taken into account in the design process. In fact, recent
earthquakes showed that in the equipments and in the contents of buildings the
damage can have large economic consequences. For instance, in high-rise
buildings, in several displacement of sensitive nonstructural systems (suspend
ceilings, light fixture, fire suppression piping systems, computer systems,
emergency power generation systems, elevators, etc.), a localized damage can
affect the functionality of a large portion of the edifice.Therefore, the structural seismic design should be applied not only to
guarantee the life safety and prevent structural collapse, but also to ensure a
given level of performance. Afterwards, the structural behavior is difficult to
analyze because a lot of uncertainty sources are involved; for this reason,
many adequate approaches have been developed. For instance, in the
probabilistic field, the methodology based on stochastic dynamic analysis is
widely adopted. In fact, Rüdinger [2] investigated the response of a nonlinear TMD subject
to a stationary white noise obtaining some optimal mechanical parameters.
Marano et al. [3–5] proposed another interesting approach based on a
procedure in which a single linear TMD problem is treated and on the fact that the
earthquake can be represented by a stationary filtered stochastic process. In
their work, the main goal is to minimize the maximum of the dimensionless peak
of displacement of the protected system in respect of the unprotected one. The
latter papers and other examples proposed in literature [6, 7] show the arising necessity of the designers to
investigate in probabilistic terms the structural response in order to face
properly the uncertainties which affect the input load. On the other hand, in a
lot of real cases, the probabilistic structural analysis is poorly suitable and
this is due to fact that a relevant number of parameters cannot be assumed as
random variables. For example, Takewaki and Ben-Haim [8] adopted some info-gap models for load and some model
uncertainty parameters in stout design to achieve the robustness function for
single degree of freedom (SDOF) and multidegree of freedom (MDOF)
structures. In the field of nonprobabilistic uncertainties, the theory based on
the concept of fuzzy sets [9] seems to be the most appropriate because of its
capability of taking into account the so-called lexical and informal
uncertainties which affect the structural system.In this paper, a new methodology is proposed to analyze the response of
a TMD subject to a nonstationary filtered white noise using a hybrid approach based
on the fuzzy and stochastic theories. Many authors dealt with fuzzy theory in analyzing
structural response [10–12] but they did not consider that also the inherent
stochastic nature of input processes can be affected by lexical and formal
uncertainties that should be treated in a fuzzy perspective. As a consequence, in
this work, all model and data parameters are assumed as fuzzy variables. So,
the membership function (MF) of the maximum standard deviation displacement of
the protected structure is gotten by means of the α-level optimization method. Moreover, an
analysis of design sensitivity is executed to establish the effects of the most
important uncertain parameters in evaluation of the structural behavior.
Finally, a fuzzy-based indicator is introduced to assess how the nominal value
of the structural performance is amplified performing a fuzzy-based analysis.


## 2. The Role of the Nonprobabilistic  Uncertainties

It should be observed that not all the unsure variables can be treated
following the standard probabilistic approach. It is not the most adequate
method to describe the uncertainty for the most part of the realistic
conditions and its use could be sometimes extremely poor for real problems. In
particular, nonprobabilistic uncertainties deal with the so-called “epistemic
uncertainty” that is seldom agreeable to probabilistic characterization. A
typical classification of epistemic uncertainty can be as follows.Idealization Uncertainty
The main goal to perform a reliable
structural analysis is the idealization of reality into a mathematical boundary
value problem. This entails continuum mechanics assumptions that provide
mathematical equations for equilibrium, kinematics, and constitutive behavior,
as well as boundary conditions. Epistemic uncertainty is thereby introduced due
to imperfect model form and/or missing parameters.Complexity Uncertainty
The parameters of an idealized
boundary value problem are commonly determined by laboratory tests of small
specimens. A useful example of this is furnished by stress paths in a
beam-column connection or the bonding between concrete and reinforcement which
are complicated phenomena whose models and data are limited. Hence, the
estimates from simplified laboratory tests carry uncertainty.Statistical Uncertainty
Parameters such as material
stiffness and strength are assessed based on available data. Uncertainty arises
when the number of observations is limited, which is frequently the case. This
leads to uncertainty in the distribution parameters of the random variables and
is referred to as statistical uncertainty. Moreover, the observations of a
variable rarely represent it perfectly and as result often a bias in data
recorded is automatically generated.From above definitions, it can be observed that nonprobabilistic uncertainties
are often present when only few data about one or more models and the structural
parameters are available or when the investigation of the properties and of the
load conditions of the mechanical and geometrical systems is economically inconvenient or
impossible. For example, this happens in the historical constructions. A
further not negligible theme is the significance of the “expert opinion” in
some phases of the design process because of the necessity and the importance
of a specific technical point of view. These observations could be summarized
in Figure1.Figure 1
The relevance of statistical data analysis.In this field, in the last few years new
methodologies has been proposed. The
most interesting one is based on the
theory of fuzzy sets [9]. Savoia [13] developed this hypothesis and defined an equivalent
and compatible probability distribution for fuzzy available data.Moreover, a fuzzy randomness application presented by Möller et al. [14] and a fuzzy Monte Carlo simulation with fuzzy adaptive importance sampling method are performed to estimate the time-dependent
fuzzy failure probability.Finally, the linear time invariant dynamical system has been
investigated by Hu et al. [15] considering that input signals are second order
fuzzy stochastic process.


## 3. Fuzzy Structural Analysis of Tuned Mass Damper

### 3.1. The Fuzzy-Based Description of Uncertainties

The variables on which lexical or informal uncertainty encumbers could be treated by
means of fuzzy sets theory. Its greatest advantage is the possibility of a
gradual assessment of the grade of membership of the elements in relation to a
generic set. This continuum degree of membership is described by the so-called
membership function (MF). Initially, in fuzzy structural analysis, one of the
problems is to acquire the MF for each fuzzy input. In this phase, the main
goal is to capture the meanings of the nonprobabilistic parameters. This problem
belongs to the area of “knowledge acquisition.” Generally, it is possible
to assert that for the so-called fuzzification, a unitary approach does not exist.
Different procedures
could be adopted for each situation relatively to the available information. In
literature, it is useful to classify the methods for constructing MF as direct or
indirect and with single or multiple experts. In our work, we assume that the
MF is known for each fuzzy variable. In other words, we suppose that a specific
and adequate knowledge acquisition approach has been performed. For a TMD, different
sources of
uncertainties can be found in both data and model. In detail, in our work, we
suppose that each variable can be modelled by means of a fuzzy number. At a
first analysis, this assumption can appear extreme but we should observe that
in any real situation it is impossible to consider a parameter as a deterministic
datum. In fact, the uncertainty can be reduced but never entirely eliminated.
In particular, in the greatest part of the circumstances, the structural
parameters are known with extreme inaccuracy. This is especially true for
structural damping. Energy absorption and dissipation as well as the structural
ductility can be burdened by a notable indeterminateness. Furthermore, the model
parameters adopted to represent a random vibration can be considered as
nonprobabilistic variables. In the dynamic analysis, a typical evidence of this
can be the valuation of the soil type; it can be stiff, medium, or soft. This
consideration is imputable to the lack of information for a reliable statistic
analysis. Finally, it is possible to notice a great variability (often in
regional and subregional scales)
of these parameters and in this case the expert opinion can play a fundamental
role in the assessment of their values and intervals of confidence. For what
concerns the TMD, we can affirm that it is an industrial product with a defect
consisting in its mechanical parameters; in fact, for them, it is often necessary
to consider a certain error in the evaluation. Besides, these devices are
frequently furnished with datasheets
in which some information on the variability of these data is indicated.


### 3.2. Theα-Level Optimization Method

Once these fuzzy
inputs are known
with their respective MF, the principal aim of a fuzzy structural analysis is
to “map” both fuzzy inputs and model characteristics into a response which is also
fuzzy. The function (or algorithm) realizing this mapping is a deterministic one
and is called mapping operator. In literature, two methodologies exist to
obtain the output MF. A first procedure is based on the concept of the extension
principle, but the most effective approach is found on the discretization of
the fuzzy sets, and so on the consequent application of theα-level optimization [16]. Let us consider some generic n fuzzy input variables:
(1)Y˜i={(yi,μ(yi))∣yi∈Yi}i=1,…,n,
where μ(yi) and Yi are, respectively, the MF and the fundamental
set of the uncertain one Y˜i (or fuzzy value y˜i). Then, for each fuzzy variable, the crisp
sets
(2)Yiαk={yi∈Yi∣μ(yi)≥αk}i=1,…,n
can be extracted for each α-cut αk∈[0,1],
and they are denoted as α-level sets. Subsequently, the Cartesian
product between the α-level sets defines the input n-dimensional crisp subspace:
(3)Y¯αk={Y1αk⊗Y2αk⋯⊗Yiαk⋯⊗Ynαk}.
If we denote with f(⋅) the deterministic mapping operator, the
following relation holds:
(4)∀u∈[minY¯αkf(y1,…,yi,…yn),maxY¯αkf(y1,…,yi,…yn)]⇒μ(u)≥αk.
Therefore,
(5)Uαk={u∈[minY¯αkf(y1,…,yi,…yn),maxY¯αkf(y1,…,yi,…yn)]}
is the output α-level set. Afterwards, the output fuzzy set
is given by
(6)U˜i=⋃α∈[0,1]Uαk={(ui,μ(ui))∣yi∈Ui}i=1,…,n.This procedure is shown in Figure 2 for n=2 and for a generic α-cut.Figure 2
Theα-level optimization.Formally, this procedure is defined by a double optimization problem for
eachα-cut, as it is expressed in (4). When the
mapping operator is nonmonotonic with reference to all the fuzzy input
variables (e.g., as it is illustrated in Figure 2), the minimum and maximum
values cannot be achieved at the vertices of the crisp subspace (3). In this
condition, a numerical algorithm must be adopted and so the procedure can
become even more complex.


## 3.1. The Fuzzy-Based Description of Uncertainties

The variables on which lexical or informal uncertainty encumbers could be treated by
means of fuzzy sets theory. Its greatest advantage is the possibility of a
gradual assessment of the grade of membership of the elements in relation to a
generic set. This continuum degree of membership is described by the so-called
membership function (MF). Initially, in fuzzy structural analysis, one of the
problems is to acquire the MF for each fuzzy input. In this phase, the main
goal is to capture the meanings of the nonprobabilistic parameters. This problem
belongs to the area of “knowledge acquisition.” Generally, it is possible
to assert that for the so-called fuzzification, a unitary approach does not exist.
Different procedures
could be adopted for each situation relatively to the available information. In
literature, it is useful to classify the methods for constructing MF as direct or
indirect and with single or multiple experts. In our work, we assume that the
MF is known for each fuzzy variable. In other words, we suppose that a specific
and adequate knowledge acquisition approach has been performed. For a TMD, different
sources of
uncertainties can be found in both data and model. In detail, in our work, we
suppose that each variable can be modelled by means of a fuzzy number. At a
first analysis, this assumption can appear extreme but we should observe that
in any real situation it is impossible to consider a parameter as a deterministic
datum. In fact, the uncertainty can be reduced but never entirely eliminated.
In particular, in the greatest part of the circumstances, the structural
parameters are known with extreme inaccuracy. This is especially true for
structural damping. Energy absorption and dissipation as well as the structural
ductility can be burdened by a notable indeterminateness. Furthermore, the model
parameters adopted to represent a random vibration can be considered as
nonprobabilistic variables. In the dynamic analysis, a typical evidence of this
can be the valuation of the soil type; it can be stiff, medium, or soft. This
consideration is imputable to the lack of information for a reliable statistic
analysis. Finally, it is possible to notice a great variability (often in
regional and subregional scales)
of these parameters and in this case the expert opinion can play a fundamental
role in the assessment of their values and intervals of confidence. For what
concerns the TMD, we can affirm that it is an industrial product with a defect
consisting in its mechanical parameters; in fact, for them, it is often necessary
to consider a certain error in the evaluation. Besides, these devices are
frequently furnished with datasheets
in which some information on the variability of these data is indicated.


## 3.2. Theα-Level Optimization Method

Once these fuzzy
inputs are known
with their respective MF, the principal aim of a fuzzy structural analysis is
to “map” both fuzzy inputs and model characteristics into a response which is also
fuzzy. The function (or algorithm) realizing this mapping is a deterministic one
and is called mapping operator. In literature, two methodologies exist to
obtain the output MF. A first procedure is based on the concept of the extension
principle, but the most effective approach is found on the discretization of
the fuzzy sets, and so on the consequent application of theα-level optimization [16]. Let us consider some generic n fuzzy input variables:
(1)Y˜i={(yi,μ(yi))∣yi∈Yi}i=1,…,n,
where μ(yi) and Yi are, respectively, the MF and the fundamental
set of the uncertain one Y˜i (or fuzzy value y˜i). Then, for each fuzzy variable, the crisp
sets
(2)Yiαk={yi∈Yi∣μ(yi)≥αk}i=1,…,n
can be extracted for each α-cut αk∈[0,1],
and they are denoted as α-level sets. Subsequently, the Cartesian
product between the α-level sets defines the input n-dimensional crisp subspace:
(3)Y¯αk={Y1αk⊗Y2αk⋯⊗Yiαk⋯⊗Ynαk}.
If we denote with f(⋅) the deterministic mapping operator, the
following relation holds:
(4)∀u∈[minY¯αkf(y1,…,yi,…yn),maxY¯αkf(y1,…,yi,…yn)]⇒μ(u)≥αk.
Therefore,
(5)Uαk={u∈[minY¯αkf(y1,…,yi,…yn),maxY¯αkf(y1,…,yi,…yn)]}
is the output α-level set. Afterwards, the output fuzzy set
is given by
(6)U˜i=⋃α∈[0,1]Uαk={(ui,μ(ui))∣yi∈Ui}i=1,…,n.This procedure is shown in Figure 2 for n=2 and for a generic α-cut.Figure 2
Theα-level optimization.Formally, this procedure is defined by a double optimization problem for
eachα-cut, as it is expressed in (4). When the
mapping operator is nonmonotonic with reference to all the fuzzy input
variables (e.g., as it is illustrated in Figure 2), the minimum and maximum
values cannot be achieved at the vertices of the crisp subspace (3). In this
condition, a numerical algorithm must be adopted and so the procedure can
become even more complex.


## 4. Dynamic Analysis of TMD under  Consideration on Fuzziness

The mechanical system is made up by a principal mass linked through a spring and a damper to another secondary mass provided with a spring and a
damper also. The latter protects the structure from high-intensity vibration
because it oscillates with the same frequency of the main structural system but
with a phase shift. The main structural system is an SDoF because in the principal
application of civil engineering we can refer to the first vibration mode of a generic
MDoF system [17]. The goal of the investigation is to get the
membership function of the peak displacement response of the described system
subject to a nonstationary seismic excitation. We can reach this aim by using
the α-level optimization procedure when all the
involved parameters are fuzzy variables.The choice of this quantity is very interesting from an engineering
point of view because it is a fundamental criterion in new design technologies
as performance- and displacement-based designs. Many authors [16] affirmed that the obtained algorithm is the best. It
is used in the optimization as an evolutionary one because it is able to
guarantee a robust optimization technique independent from the OF type and
capable of finding some global optima. For these reasons, the use of a genetic
algorithm seems the best preference to adopt. In all conducted simulations, a
standard MATLAB genetic algorithm has been used [18].The dynamic mechanical model and its properties are reported in Figure3. Hence, for the sake of simplicity, the symbol αk is omitted for each fuzzy parameter.The structural dynamical model.Being(7)μT=⋃α∈[0,1]μTαk=⋃α∈[0,1](mTαkmsαk).
If D¯αk denotes the input 8-dimensional crisp
subspace, given by the following Cartesian product:
(8)D̲αk=[Ωsαk⊗Ξsαk⊗ΩTαk⊗ΞTαk⊗MTαk⊗Ωfαk⊗Ξfαk⊗X¨gmax,αk],
the standard deviation of the response is given by
(9)Σαk={σXmax∈[minD¯αkf(ωS,ξs,ωT,ξT,ωf,ξf,μT,X¨gmax),maxD¯αkf(ωS,ξs,ωT,ξT,ωf,ξf,μT,X¨gmax)]},
where in this particular case the mapping operator is provided by
(10)σXmax=f(ωS,ξs,ωT,ξT,ωf,ξf,μT,X¨gmax),
so that, the fuzzy output set is calculated by
(11)Σ˜=⋃α∈[0,1]Σαk={(σXmax,μ(σXmax)∣σXmax∈Σ}.
Our purpose is to find Σ˜ by the mapping operator defined in (10). We can say
about the input process that the response evaluation must be based on a
stochastic approach because of its random nature. As commonly known, the ground
accelerograms are often modelled as a zero-mean stochastic nonstationary
process. This assumption is able to consider the intrinsic probabilistic nature
of earthquakes in the real way. Moreover, a suitable description with no
constant contents both in amplitude and in frequency could be taken into
account. A widely
used stochastic approach is the one proposed by Tajimi [19], based on a linear second-order filter applied to a
stationary white noise. Therefore, the ground acceleration X¨g is calculated by
(12)X¨g(t)=X¨f(t)+φ(t)W(t),X¨f(t)+2ξfωfX˙f(t)+ωf2Xf(t)=−φ(t)W(t),
where Xf(t) is the response of the Kanai-Tajimi filter,
having frequency ωf, damping coefficient ξf,
eventually time depending, and white noise W(t) whose constant bilateral power spectral density
(PSD) function is S0.
It is related to the peak ground acceleration (PGA) X¨gmax by means of relation [20]
(13)S0=0.222πξf(X¨gmax)2ωf(1+4ξf2).
No stationary characteristic is introduced by the deterministic temporal
modulation function. It controls the intensity variation without changing the contents
of the earthquake frequency. Moreover, different modulation functions have been
proposed in literature with the aim of best accordance with the real-time
evolution of accelerograms. In our work, we adopt the modulation function
proposed by Jennings et al. [21] as follows:
(14)φ(t)={(tt1)2,t<t1,1,t1≤t≤t2,e−β(t−t2),t>t2.
Making the assumption that the system is linear and the input is
Gaussian zero-mean process, the knowledge of the covariance matrix can fully
characterize the space state response statistics, as a result of the fact that
the response is Gaussian zero-mean too. Then, the probabilistic analysis is performed
by solving the well-known Lyapunov matrix differential equation.Therefore, let us consider the state vectorZ¯:
(15)Z¯(t)=[Xs(t)XT(t)Xf(t)X˙s(t)X˙T(t)X˙f(t)],
the motion equation of the structure plus filter in the space state is
(16)Z˙¯(t)=AZ¯(t)+F¯(t),
where
(17)F¯(t)=(00000−φ(t)W(t))
is the forcing vector, and
(18)A=(000100000010000001𝒰μTωT2ωf2𝒫2μTξTωT2ωfξfωT2−ωT2ωf22ξTωT−2μTξTωT2ωfξf00−ωf200−2ωfξf)
(where 𝒰 denotes −ωS2−μTωT2 and 𝒫 denotes −2ξSωS−2μTξTωT) represents the system matrix.So starting from the state equation (16), the covariance analysis
is performed by solving the Lyapunov matrix differential equation:
(19)R˙Z¯Z¯(t)=ARZ¯Z¯(t)+RZ¯Z¯(t)AT+B(t).
In (19),
(20)RZ¯Z¯(t)=E[Z¯Z¯T]
is the covariance matrix (E[⋅] denotes the mathematical expectation),
while B(t) contains all elements equal to zero except the
last one that holds 2πS0φ(t)2.Afterwards, if we consider the time interval[0,T],
the relevant quantity can be obtainable by
(21)σXmax=f(ωS,ξs,ωT,ξT,ωf,ξf,μT,X¨gmax)=maxt∈[0,T]RZ¯Z¯(1,1).
For a generic αk,
expression (9) becomes
(22)Σαk={σXmax∈[minD¯αkmaxt∈[0,T]RZ¯Z¯(1,1),maxD¯αkmaxt∈[0,T]RZ¯Z¯(1,1)]}αk.


## 5. Numerical Examples

In the present work, several numerical simulations have been carried out
to assess the proposed methodology and to conduct a sensitivity analysis. For
all the simulations, all the membership functions of fuzzy variables are
supposed to be triangular and the amplitude of the modulation function is 7 seconds
in (14), while β=0.4.
The α-cut step is fixed to 0.2.The first analysis has been conducted to calculate the membership
function ofσXmax assigned the values reported in Table 1.Table 1
Fuzzy input for the numerical example.
Fuzzy variablesValuesω˜f〈10,15,20〉[rad/s]ξ˜f〈0.35,0.45,0.65〉[ad]ω˜s〈10,12,15〉[rad/s]ξ˜s〈0.01,0.02,0.05〉[ad]X¨˜gmax〈0.15,0.20,0.25〉[g]μ˜TMD〈0.04,0.08,0.12〉[ad]ω˜TMD〈9,12,14〉[rad/s]ξ˜TMD〈0.08,0.12,0.15〉[ad]About the filter frequency and the damping ratio, the support describes
the values that a stiff soil can assume with a medium assessment given by the
core. In practical applications or in research works, the estimation of the soil
type is affected by a remarkable degree of uncertainty which can be difficult
quantified. Moreover, in the prefilters technique which is an extensively
adopted methodology to model the ground in dynamic structural analysis, the
evaluation of the values of the parameters is often purely conventional and
there are no physical reasons to justify them once they have been defined. In
fact, in literature, there are many authors’ proposals about this issue [22] but none of them seems to be coherent to design code
spectra or to scale really seismic simulations. For what concerns the support
assigned to the structural frequency and damping ratio, we can say that the
values are typical of civil buildings. In practical applications, their
assessment is difficult because there are few experimental data about them. The
best analysis which can be performed is a postevent one but it is worthless. As
a consequence, the properties of the real structure are often subject to the expert
judgments which are evidently affected by informal and lexical uncertainties. Another
important aspect is the ground acceleration support which represents the values
that can be assumed in evaluating a site classified as B by the Eurocode 8. It
is well known that the hazard maps expressed in terms of peak ground
acceleration or in general by means of an intensity factor are notably
simplified; in fact, real analysis on the behavior of the seismic waves show
that they change both their characteristics and their energy content going
through each soil layer. As a consequence, the spatial variation of the peak
ground acceleration is a significant source of uncertainty. The values of the TMD
parameters are regularly presented in the datasheets of the producers but for
each of them a certain degree of tolerance has to be considered. Simulation
outcome is shown in Figure 4. Examining Figure 4, several considerations can be made distinguishing
several values of degree of membership.Figure 4
The maximum displacement standard deviation MF.Ifμ(σXmax)=60÷100%,
the output values are evidently concentrated near the core. So if we assume (as
widely adopted in current literature) that the core corresponds to the probabilistic
mean value, this denotes that the confidence interval of structural response
estimation is near the one that can be obtained by performing a purely
probabilistic analysis.Ifμ(σXmax)=30÷60% (so the degree of membership is considerably
lower), it can be noticed that the distance of the output values from the core
is increasing. In practical, occurrences these values of response degree of membership, which
cannot be usually excluded, lead
to a lower assessment of the structural performance and to an overestimation of
the structural reliability.Ifμ(σXmax)=0÷30%,
the latter trend continues; in fact, the scattering from the core is very huge.
However, we have to mention that this degree of membership can rarely occur in
real applications.Once the MF of response has been obtained, a sensitivity analysis has
been conducted to affirm how a scattering of both structural frequency and
damping can affect its behavior. To complete these investigations, two types of
MF forω˜s and ξ˜s are considered. One with a core moved to the
left of MF and one moved to the right. The numerical values are represented in Tables 2 and 3, respectively, for ω˜s and ξ˜s.Table 2
Different adopted structural frequency membership functions.
Fuzzy variablesValuesω˜S〈10,10.50,15〉[rad/s]ω˜S〈10,14.50,15〉[rad/s]Table 3
Different adopted structural frequency membership functions.
Fuzzy variablesValuesξ˜S〈0.01,0.0125,0.05〉[ad]ξ˜S〈0.01,0.0475,0.05〉[ad]Figures5 and 6 show the result of the conducted sensitivity analysis
with respect to ω˜s and ξ˜s.
A first observation is evident. Examining the output MF in the first case (blue
line), we can notice that the core has a value which tends to be zero because
moving ωs to the right is equivalent to supposing that the system is very stiff, the same holds
for ξ˜s,
moving this parameter MF implies to ensure a high-structural dissipation
capacity. In addition, we have to mention that the blue and the red lines are
almost superposed for what concerns the left branch of the MFs. Moreover, in Figure 6 the cores of the two MFs are very near to each other. However, in both cases the
difference of structural response tends to be against safety considering only
the right branches of the MFs. A variation in the structural frequency causes
the highest scattering in the structural response.Figure 5
The sensitivity analysis on the structural frequency.Figure 6
The sensitivity analysis on the structural damping.Finally, a fuzzy indicator has been derived to estimate how much the
defuzzificated output MF is far from the nominal output valueσcXmax at μ(σXmax)=1.
The defuzzificated variable is gotten evaluating the center of gravity of the
fuzzy number Σ˜:
(23)G(Σ˜)=∫σlXmaxσuXmaxσXμ(σX)dσX∫σlXmaxσuXmaxμ(σX)dσX,
where in (23) the subscripts “u, 1, and c,” respectively, stay for upper, lower,
and core. The meaning of these subscripts is graphically represented in Figure 7 in order to clarify how structural amplification factor is
obtained.Figure 7
Generic representation of a membership function with representation of
symbols used in (23).The above-mentioned fuzzy indicator, named, structural response
amplification factor is given by(24)γ=G(Σ˜)σcXmax.Applying (23) to the MF
given by Figure 4, we obtain a value of the centroid of 13.60 cm, while the
value of σcXmax is equal to 1.51 cm; afterwards γ=9.
This result leads to several considerations. First of all, σcXmax is determined performing a probabilistic
analysis based on the consideration that the structural data are deterministic,
and so assuming that the knowledge of the parameters is complete. Instead, if
the parameters are known with a certain degree of uncertainty, a mean estimate
of the response is given by the centroid value which is one order greater than
the nominal value. However, it has to be underlined that this evaluation corresponds
to a low degree of membership which rarely occurs in the practical
applications. Then the
coefficient γ is a measure of how much the probabilistic
response value is amplified if an average degree of informal uncertainty is
assumed in the values of the parameters.


## 6. Conclusions

The proposed approach has been adopted to evaluate the structural
response of a linear TMD subject to an earthquake input modelled as the Kanai and Tajimi filtered
nonstationary process. In the space state, the dynamical analysis has
been conducted considering that all the involved structural and filter
parameters are fuzzy sets; this choice has been done to consider that in real
analyses all of them are affected by a degree of uncertainty which cannot be
quantified only using a probabilistic approach. In fact, it is necessary to
take into account the uncertainties also. The fuzzy structural analysis has
been performed by means of theα-optmization method. A genetic algorithm has
been used to gain the optimal point which is able to define the output MF under
the hypothesis that the MFs of the input variables are all triangular. The results
show the effectiveness of the numerical proposed procedure, demonstrating that
assuming a high-medium level of confidence and considering that the core of the
output MF corresponds to the probabilistic mean value, the estimation of the structural
response is almost the same if the purely probabilistic structural analysis is
performed. Afterwards, a sensitivity analysis in respect to the structural
frequency and damping ratio has been conducted. The results of this analysis
show that a scattering in ω˜s has more influence in the output MF. So,
judging inopportunely these parameters without considering in a proper way the
degree of uncertainty as using wrong values for them could induce an erroneous
assessment of the structural response. Finally, a structural amplification
factor is obtained by dividing the centroid value of a fuzzy output number and of
the core one. When the amplification factor is greater than one, the structural
response is overrated regarding the stochastic approach. This clearly shows
that the fact of performing a stochastic analysis of a structural system and of
considering a certain degree of uncertainty on its parameters is often not
sufficient to properly evaluate the structural response and this could induce an
erroneous assessment of its values. Frequently, this scattering can be also
very huge, so the analysis process cannot be properly controlled.



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*Source: 207254-2009-04-06.xml*