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7/29/2019 2012-12-26-Symptom_based Reliability and Generalizedre Pairing Cost
1/9
Symptom-based reliability and generalized repairing cost
in monitored bridges
R. Ceravolo , M. Pescatore, A. De Stefano
Politecnico di Torino, Torino, Italy
a r t i c l e i n f o
Article history:
Received 24 May 2007
Received in revised form
22 January 2009
Accepted 11 February 2009Available online 20 February 2009
Keywords:
Structural reliability
Safety assessment
Modal testing
Health monitoring
Symptom
Generalized maintenance cost
a b s t r a c t
This paper proposes the use of structural safety formulations conceived to take into account the
presence of periodic monitoring systems. Monitoring is a valid tool to improve the safety of those
structural systems that cannot withstand invasive tests or interventions that would alter their nature or
their intended use. Reliability can be defined as a function of a measurable quantity that reflects the
damage, referred to as symptom, and it can also be defined as a function of several symptoms
considered simultaneously. A knowledge of the current value of a symptom makes it possible to
determine the residual damage capacity and the residual lifetime of a structure. Redefining structural
safety in terms of residual lifetime provides the theoretical framework for the introduction of vibration-
based monitoring activities in probabilistic formulations. In the last part of the paper, by relating
damage to reliability with respect to collapse, the generalized maintenance cost for a concrete bridge
deck was analyzed in order to verify the economic advantages offered by dynamic monitoring.
& 2009 Elsevier Ltd. All rights reserved.
1. Introduction
According to the probabilistic methods used most widely for
the assessment of safety in the structural field, reliability is
defined as the probability of the structure attaining a limit state
during a predetermined period of time. In many instances, this
assessment is summarised by an ad hoc reliability index [1].
Albeit useful at the design stage, this approach displays some
limitations when dealing with existing structures:
it does not take into account the additional knowledgeobtained from monitoring activities;
it often overlooks the fact that safety deteriorates over time; it does not provide the information needed for a long-term
evaluation of the economic convenience of restoration works.
In particular, the latter task calls for a knowledge of the residual
lifetime or service time of a structure, which must be a factor in
the assessment of the overall economic utility of a strengthening
intervention.
New studies on reliability-based reassessment of structures
focused on updating the probability of failure according to new
information coming from existing structures [24]. Probability
models may thereafter be enhanced by collecting new data
regarding geometry, material properties, structural deterioration,loading on the structure, static and dynamic behaviour [2].
If the degradation of reliability over time is taken into account,
the lifetime of a structure is a random variable [5] and reliability
can be characterised in relation to the so-called hazard function
(the damage rate in the infinitesimal time interval), which
assumes various forms depending on the distribution model
adopted (Weibull, Gamma, Frechet, etc.). In this connection, a
monitoring-oriented approach [6] is of great interest, as the
monitoring process is able to supply useful data both to plot the
reliability curves, defined as a function of the symptom, and to
interpret the diagrams obtained.
Structural monitoring, construed as a system that provides on
request data regarding a specific change, or damage, occurring in a
structure, can be a valid tool to fine-tune reliability estimates inthe light of the actual conditions of a structure. With structural
monitoring systems reference is understood to devices (hardware)
and procedures (software) used to acquire the time evolution of
parameters that are supposed to be related to the safety condition
of a structure (usually strains, displacements, velocities, accelera-
tions, temperatures, forces). Today, the trend is to consider
information coming from monitoring systems as crucial to
decisions about retrofitting existing structures [7]. Recently,
research focused on the possibility of obtaining more information
on the safety condition of a structure on the base of vibration
measurements [8,9]. The basic idea behind current dynamic
monitoring techniques is that modal parameters are a function
of the physical properties of the structure; therefore, changes
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Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ress
Reliability Engineering and System Safety
0951-8320/$- see front matter & 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ress.2009.02.010
Corresponding author.
E-mail address: [email protected] (R. Ceravolo).
Reliability Engineering and System Safety 94 (2009) 13311339
http://www.sciencedirect.com/science/journal/magmahttp://localhost/var/www/apps/conversion/tmp/scratch_5/http://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.ress.2009.02.010mailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_5/dx.doi.org/10.1016/j.ress.2009.02.010http://localhost/var/www/apps/conversion/tmp/scratch_5/http://www.sciencedirect.com/science/journal/magma7/29/2019 2012-12-26-Symptom_based Reliability and Generalizedre Pairing Cost
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in the physical properties will cause detectable changes in
modal properties. The advantage of this approach is that a local
measurement can provide information related to the global
behaviour.
In this paper, the symptom-based approach is analyzed, in
order to evaluate its applicability to vibration-based structural
monitoring. In the last part of the paper, also based on reliability
with respect to the ultimate limit state (ULS) of concrete bridges, acost analysis for concrete bridge decks was performed in order to
verify the economic advantages offered by dynamic monitoring.
2. Symptom-based approach to safety assessment
Let us now examine a symptomatic approach to the evaluation
of the performance of a structure, so that reliability appreciation
depends on measurable quantities. It is first assumed that when a
symptom exceeds an assigned value, Sl, the structure does not
fulfil the requirements for which it has been designed, and the
unit is definitely in need of repair or replacement [6]. In practice,
an excessive value of the symptom (e.g. the deflection of a bridge
or, as in our case, its fundamental period) would result inthe structure being excluded from the monitoring program. If
the reliability of a structure, R(t), is defined as the probability that
the time it takes a system to reach a damage limit state associated
to the structures lifetime, tb, is greater than a generic time t:
Rt Ptptb, (1)
then reliability can be rewritten as a function of the symptom
variable, S; in this case, it is defined as the probability that a
system, which is still able to meet the requirements for which it
has been designed (SoSl), is active and displays a value of S
smaller than Sb, where Sb is the value of the symptom
corresponding to the reference limit state. Accordingly, reliability
is defined as
RS PSpSbjSoSl
Z1S
fSdS, (2)
i.e., R(S) can be expressed by the integral of the symptoms
distribution probability density fS. With the symptomatic ap-
proach it is also possible to work out, for the R(S) function,
expressions similar to those used by the time-based approach,
that is to say for R(t); the hazard function, h(t), specifies the
instantaneous rate of reliability deterioration during the infinite-
simal time interval, Dt, assuming that integrity is guaranteed up
to time t [5]:
ht limDt!0
PtptbotDtjtbXt
Dt. (3)
h(t) is connected to the reliability function, R(t), by the following
relationship:
Rt exp
Zt0
ht0dt0
. (4)
In a similar manner, the so-called symptom hazard function,
h(S), is defined as the reliability deterioration rate per unit of
increment of the symptom:
hS limS!0
PSpSboSDSjSbXS
DS, (5)
hence
RS exp ZS
0hS0 dS0
!(6)
or equivalently [5]
hS 1
RS
d
dSRS. (7)
Iftb is the time of attainment of a damage limit state or the total
lifetime, reliability as a function of the symptom gives the residual
damage capacity, DD, of the structure:
RS 1 DS DDS, (8)
where D t(S)/tb represents the systems aging as well as the
measure of the damage. In practical applications, symptom
models are chosen that lead to realistic expressions for the
residual damage capacity. For instance in structural systems,
which are characterized by aging or failure processes, models with
increasing hazard functions (exponential, Weibull, gamma, log-
normal distributions, etc.), are used the most [5].
Eq. (8) lends itself to a diagnostic use: assuming that one knows
the evolution of reliability through the observation of a set of
systems, the value of the symptom as observed in a given unit makes
it possible to determine the residual lifetime of the unit itself.
Under this approach monitoring plays a key role, in that
reliability is no longer expressed as a function of time, but rather
as a function of a symptom, which is a measurable quantity.
3. Extension to structural classes
Reliability can be described starting from a primary reliability,
R0(S), that applies to a given type of systems (structural class)
and can be characterised for a particular system by the introduc-
tion of a logistic vector Li, with i 1yN, where N is the total
number of systems to be monitored [10]. Li denotes the individual
element of the sample, it may contain a series of specific
parameters depending on which aspect of the system we know
or we want to monitor.
Each unit of the class may differ from the other elements in its
original characteristics as well as its usage (e.g. actions ormaintenance quality). Any additional information or measure-
ments, directly or indirectly related to the symptom S, i s a
potential component of the logistic vector, as long as it is referred
to the single unit. In principle this may be geometry, loads,
environment parameters, material properties, soil, maintenance
levels, other factors even of a binary nature.
The Li vector appears in the formulations of system reliability
starting from h(S), which depends on L:
hSL hS; L, (9)
whence, by integration and by analogy with Eq. (5), we can
express the value of reliability, R(S,L), as a function of the
symptom considered and the L vector:
RS; L exp
ZS0
hS0; LdS0( )
. (10)
For the hazard function, we start from a general multiplicative
form of the primary hazard function:
hS; L h0SgL (11)
in order to explore how the logistic vector, L, affects the survival
function R(S,L). In the assumption of small changes of L, system
reliability can be determined as [10]
RS; LjL0 DL R0S; L0 1 DLTqg
qLlnR0S; L0
& '(12)
being : R0S; L0 exp ZS
0h0S0gL0 dS0
( ), (13)
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where g(L) is any function satisfying the condition g(L0) 1, and
DL is a vector containing the variations made to the parameters of
the L vector. When g(L) is assumed to be linear Eq. (11) tends to a
Cox proportional model [11].
4. Interaction between monitoring and repairing costs
In order to analyze the interaction between structural
reliability and maintenance costs, directed to restore the initial
reliability of the structure, in the following we shall refer to the
ultimate limit state associated with structural failure, as related to
a limit state of damage (DLS) directly correlated to the structures
lifetime.
Though a full reliability approach is a possible option, in
accordance with previous works in the field of structural
engineering, it is assumed that safety at the ultimate limit state,
as expressed by index bULS F1(1RULS), where F
1 is the
inverse of the standard normal cumulative function [1], is
indirectly correlated to the residual life of the structure. The
following expression is thus used for the cost Ci of a generic
maintenance intervention [12]:
Ci fb0ti;Db0ti, (14)
where Ci is the cost of the ith intervention, b0(ti) is the reliability
of the structure at the time of the intervention performed at time
ti, Db0(ti) is the increase in reliability caused by the intervention.
Hence, in addition to depending on type of intervention (whose
effect is Db0(ti)), the variability of maintenance costs is also
affected by the health conditions of the structure at the time
maintenance works are performed; in general, structures in good
conditions require lesser amounts compared to rundown struc-
tures, the reliability level attained and the type of intervention
being the same.
In particular, the cost function C1, used in this analysis and
relating to a single maintenance intervention applied at time t1,
includes a fixed part, C0, which does not depend on theimprovement achieved with the intervention, and a variable part,
which is a function ofDb0(t1) [12]:
C1 C0 lsDb0t1q, (15)
where s and q are cost parameters, l is a multiplication factor that
takes into account the reliability level b0(t1) of the structure at the
time the intervention is performed.
The increase Db0(t1) in reliability b0(t1) obtained with the
maintenance intervention is designed to restore the reliability of
the structure to the initial value of the reliability index. It is
assumed that the value ofDb0(t1), induced by the maintenance
intervention performed at time t1, can vary beginning from a
threshold value that has to be assured anyway in case of
intervention. The multiplier l is determined as follows [12]:
l p1b0t12 p2b0t1 p3, (16)
where p1, p2 and p3 are coefficients that vary as a function of type
of intervention.
Due to the maintenance intervention at time t1, the reliability
profile is updated as follows:
bt; t1 b0t; t0ptot1;
b0t Dbt1; t1pt;
((17)
where b0(t) is the profile of the reliability index before the
intervention, Db(t1) is the increase in the reliability index caused
by the maintenance intervention, and b(t,t1) is the evolution of
reliability following the intervention applied at t1.
Another factor to be considered in addition to the maintenancecost, is the failure risk cost Cpf1;b0
to be evaluated from profile
b(t,t1) as modified, with respect to the virgin index, b0(t), by the
intervention performed at t1 [12]:
Cpf1;b0
cfth
Ztht0
Dbt; t12
1 vtdt, (18)
where th is the time period encompassed by the analysis of costs,
Db(t,t1) is the deviation of b(t,t1) from the value of b0 at initial
time t0, Db(t,t1) b0(t0)b(t,t1) if b0(t0)Xb(t,t1), otherwiseDb(t,t1) 0; cf is the coefficient reflecting the risk cost, v is the
discount rate of money. The magnitude of cf depends on several
factors, such as type of structure, the volume of daily traffic, the
number of accidents, and service disruption.
The total cost function CTOT, depending on the maintenance
intervention time t1, is the sum of the maintenance intervention
cost C1 (Eq. (15)) and of the risk cost Cpf1;b0:
CTOTt1 C1t1 Cpf1;b0t1. (19)
Let us now consider the evaluation of the economic conve-
nience of monitoring the generic structure. If the reliability profile
b(t) of the bridge deviates at time t1 by DDb(t1) from b0(t), i.e., the
value that applies to the entire class of structures, the main-
tenance interventions, selected on the basis of the values of curveb0(t), will not be able to restore b(t1) to the initial value b0(t0), and
will only obtain a lower value: b0(t0)DDb(t1). The advantage
offered by the monitoring process, and hence by a correct
knowledge of the actual profile, b(t), compared to the standard
one, b0(t), makes it possible, with an additional maintenance cost
incurred to restore b(t1) to b0(t0), to avoid the risk cost associated
with DDb(t1) during the time following the intervention (Fig. 1).
The expression for the determination of the failure risk cost,
Cpf1;b , that applies to a generic evolution b(t) (other than b0(t)) and
to the maintenance intervention at time t1, becomes:
Cpf1;b Cpf1;b0 cfth
Ztht1
DDbt12 2DDbt1Dbt; t1
1 vtdt
cfth
Zt1t0
DDbt2
2DDbtDbt; t1
1 vtdt, (20)
where from the risk cost Cpf1;b0for the standard reliability profile,
b0(t), we subtract the risk cost for the time t1th, avoided thanks to
the monitoring and we add the additional risk cost for the time
t0t1. DDb(t) is the deviation of the monitored reliability profile
b(t) from the standard reliability profile b0(t). The total cost
function CTOT, depending on the maintenance intervention time t1,
becomes:
CTOTt1 C1t1 Cpf1;b t1 Caddt1. (21)
Besides the maintenance intervention cost C1, the cost of the
additional maintenance Cadd is included in Eq. (21) and is obtained
by the following formula (22), that is analogous to Eq. (15):
Cadd lsDDbt1q. (22)
Whenever relevant, also the cost of monitoring may be included
in Eq. (21).
5. Dynamic monitoring of bridge decks
The application example proposed below (Fig. 2) uses simply
rested prestressed bridge beams (95 m span, box section, fck 40
N/mm2, fctm 3.5N/mm2, elastic modulus Ecm of concrete 35
kN/mm2, area of the cross-section Ac 11.53 m2, and moment of
inertia JG 23.086m4).
The symptom that we assume to monitor over time, through
customary experimental modal analysis procedures is the funda-mental period, T, of this structural class, whose variation is
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associated with the decrease in stiffness. Due to the purely
methodological value of this example, the symptoms evolution
is not measured but calculated according to an analytical
damage model. The first cracking moment of the deck, Mcr,
is 53MNm, as determined according to the following
formula [13]:
Mcr fctmWi, (23)
where fctm 3.5N/mm2 is the average tensile strength of concrete
and Wi is the section modulus of the uncracked section at the
lower chord.
Having defined a log-normal statistical distribution of the live
load q on the bridges referred to a 1-year period (mean value:
45 kN/m, variation coefficient: 0.15), the analysis is performed for
each time step and each load level envisaged [1,13,14]. The spatial
distribution of the loads was performed according to the rules set
forth in the European standards [13,14]. Another option would be
to turn to bimodal load distributions [15], but this is far beyond
the scope of the present paper.
The damage model considered is based on the elastic theory of
damage [16], so that the deterioration process in the bridges,
triggered when the damage threshold envisaged was exceeded,translates into a reduction in bending stiffness, EI, according to the
following expression [17]:
EI EI01 d, (24)
where EI0 is the stiffness of the uncraked section, and d is the
damage parameter. The isotropic damage parameter d can be
physically interpreted as the ratio of damaged surface area,
corrected by stress concentration effects and interactions, over
total surface area at a local material element [18].
The deformation energy, corresponding to the first cracking
moment, Mcr, is assumed as the threshold value, x0, beyond which
the damage mechanism is triggered in the beam.
For a given time step, each statistical value of the load q
corresponds to an accumulated deformation energy, x, and a certain
size of the crack zone in the beam astride its midspan. The damage
to the deck, reflected by parameter d, affects only the cracked zone
and propagates over time according to the elastic model: at the n+1
interaction, the elastic deformation energy, xn1n1, a function ofthe state of strain, en+1, is determined; we get the damage parameter,dn+1, and the damage threshold, rn+1, as given below [16]:
dn1 dn if xn1orn;
1 1 Ax0=xn1 A expBx0 xn1;(rn1 maxrn;xn1. (25)
ARTICLE IN PRESS
(t)
(t)
0(t)
Additional failure safety
deterioration from monitoring
1
1
Failure safety deterioration for the standard
reliability profile
Failure safety advantage from
monitoring
Standard reliability
profile
Reliablity profile obtained by
structure monitoring
t1t
Fig. 1. Advantage offered by structural monitoring when the reliability profile is lower than the standard reliability profile. t1 is the time of the maintenance intervention.
1270 cm
40
335250
45
510
70
45335
Fig. 2. Bridge deck section. Measurements are in centimeters.
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In Eq. (25), the deformation energy, xn+1, is compared with the
limit, rn, and the damage parameter, dn+1, is maintained the same as
in the previous step if the accumulated energy, xn+1, does not exceed
the limit rn; conversely, dn+1 is defined with Eq. (25) ifxn+1 exceeds
rn. In Eq. (25), A and B stand for the growth coefficients of the
damage law, which are 0.68 and 1.41, respectively, for high strength
concrete [18]. The fundamental period was calculated by averaging
over simulated outcomes referred to a single year, as resulting fromMonte Carlo simulations based on the distribution assumed for
loads.
It should be noted that with the build-up of the damage
resulting in the decrease in stiffness, EI, the fundamental period T
of the structure increases asymptotically up to a value Tf of about
1.25 s, with a 14% increment over the initial value of the period,
T(t 0), which was 1.09 s.
The residual damage capacity of the system, or its reliability as
a function of the symptom observed, R(S), is given by [6]
RS DDS 1 tS
tb. (26)
In this application it was assumed for the bridges lifetime
tb 100 years.In evaluating the reliability of existing structures one cannot
rely on monitoring data covering the entire life of a structure, on
the other hand it has been ascertained that a few initial data
regarding the symptom observed over time are sufficient to
identify the underlying trend evidenced by it. The symptom-based
approach makes it possible to choose, from among different
variation curves of the symptom over time, the one that best
reflects the trend observed so as to obtain a tool for the evaluation
of the current and future conditions of the system.
By way of exemplification, the evolution over time of the
symptom/fundamental period can be approximated with a
lifetime distribution model [5] (Fig. 3a). For instance, in this case
a possible option would be a Weibull model:
S=St0 1 1a ln1 t=tb
1=g (27)
with the coefficients a 4.6 and g 6.2. Correspondingly thereliability (Fig. 3b) would become:
RS expfSSt0 1a
gg, (28)
whose associated hazard function h(S) is monotone increasing
(g41) with the symptom (Fig. 4) [5]. Apparently, in this example,the Weibull and the Frechet models tend to overestimate
reliability in the short/medium period, while they are conserva-
tive when the bridges service time is approaching tb. Obviously,
the selection of a specific model will depend on the monitoring
experiences performed on different structural types.
Knowing the current value ofS, Eq. (28) supplies an evaluation
of the current and future conditions of the structural class in
terms of residual lifetime or primary reliability.
Given the primary reliability function, R0(S,L0), valid for a
family of structures of the same type, by monitoring a single unit
in the class it is possible to calibrate with greater accuracy theestimate of its residual damage capacity, R(S,L).
If monitoring results reveal an evolution of the symptom faster,
or slower, than the standard rate assumed for the structural
family, the estimate for the specific structure in question can be
modified through Eq. (12), where R0(S,L0) is the survival function
for the standard structure, and R(S,L)L0+DL is the survival function
for a specific structure characterised by increment DL of the basic
logistic vector, L0.
If R0(S,L0) is made to coincide with R0(S) and the measured
fundamental period Tm is the only monitored quantity to be
inserted in the logistic vector, the following form may be assumed
for Eq. (11):
hS; L h0SgL h0SL h0ST
mT0 . (29)
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1.161.14
1.12
1.1
1.08
1.06
1.04
1.02
T/T(t=0)
0
1
0.02 0.04 0.06 0.08 0.1
t / tb
Elastic theory of damage
Exponential type distribution model
Weibull distribution model
Frechet distribution model
1
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.9
R
1 1.05 1.1 1.15
T / T (t =0)
Fig. 3. (a) Symptom evolution by the elastic theory of damage and by statistical models. (b) Damage limit state reliability as a function of the symptom.
12
10
8
6
4
2
0
Sym
to
mhazard
func
tion
1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26
Fundamental period T
Fig. 4. Symptom hazard function h(S) resulting from a Weibull model. In this case
the symptom is the bridge decks fundamental period T.
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In other words, hazard function h(S,L) is assumed to be modified
proportionally to the measured symptom Tm referred to its
primary value, T0. For generalitys sake, in practical applications
Tm and T0 may be conveniently referred to their initial values.Correspondingly Eq. (12) becomes
RS; LjL0 DL R0S 1 DT
T0lnR0S
& ', (30)
where a positive deviation in the symptom, DT TmT0, indicates
a reduction in reliability. The evolution of the fundamental period
of the structure in time is represented in Fig. 5a, while the graphs
of RDLS(t), corresponding to different evolutions of natural period,
are shown in Fig. 5b.
6. Interaction between reliability and costs
A cost analysis, according to Section 4, has been applied to thebridge deck. For the sake of simplicity, here it is assumed that
reliability with respect to failure (RULS) is indirectly related to
reliability with respect to damage (RDLS), which governs the
structures residual lifetime.
The reliability index profiles, bDLS (Fig. 5d) have been obtainedfrom RDLS through the following relationship:
bDLSt F11 RDLSt. (31)
Likewise the graphs ofbULS(t) (Fig. 5c), have been obtained from
the reliability RULS. Curves in Fig. 5c refer to different values ofDT/
T0 (0.1, 0.2, 0.5, respectively), virtually found with monitoring.
The evolution of the total cost, including the maintenance cost
and the risk cost, has been obtained through the formulas (20)
and (21) as a function of the intervention time (Fig. 6). The
adopted values for the cost parameters associated to the chosen
type of intervention are indicated in Table 1.
If the reliability profile bULS(t) is lower than the standard one,
bULS,0(t), the monitoring process results to be advantageous from
the economic standpoint, as long as the risk cost avoided exceedsthe additional maintenance cost.
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1.16
1.14
1.12
1.1
1.08
1.06
1.04
1.02
1
T/T(t=
0)
0 5 10 15 20
Time (years)
L = T/T0= 0.5
Elastic theory of damage
Monitored symptom profiles L >0
Monitored symptom profiles L
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Then the graph ofbULS(t) has been approximated through the
function [12]:
bULSt bULSt0 at t00:5, (32)
where the degradation parameter a0 is assumed to be 0.109 for thestandard profile bULS,0 (Fig.1) and bULS(t0) is assumed to be 5.5. If a
uniform probability distribution is assumed for a parameter [12](see Fig. 7b), different variation coefficient, c
a, for the statistical
distribution produce the graphs shown in Fig. 7a: the higher the
variation coefficient the more advantageous the effects ofstructural monitoring. Correspondingly the maintenance inter-
ventions result to be slightly anticipated with monitoring.
7. Modal testing: sensitivity of different parameters
The field of structural identification now offers a vast range of
effective techniques. In the civil engineering field, of special
interest are methods which do not require a prior knowledge of
the dynamic input and are able to take advantage of the natural
excitation to which a structure is subjected, so as to enable the
behaviour of the structure to be monitored in operating condi-
tions [19]. In recent years, time domain techniques have been
used rather successfully [20], thanks to the great spectralresolution offered and to their modal uncoupling capability.
The situation is more critical for damping estimation, since this
parameter, having no significant effects on frequency, primarily
affects the modulation of modal signals and, in unknown input
conditions, becomes latent information. In actual fact, the
accuracy in damping estimation afforded by current output-
only methods is not very high [21]. These considerations
prompted some proposals for timefrequency methods, which
are able to handle non-stationary excitation typical of bridges and
other civil structures [22], but, at the same time, bring about
complexity and computational cost. We conclude that, while
modal frequencies may be evaluated efficiently through standard
output-only identification procedures, damping monitoring in
civil structures still requires the excitation to be measured and
this may prove costly. This notwithstanding, in the following we
present an example in which both frequency and damping havebeen ideally monitored.
The numerical application described below is about reinforced
concrete bridge piers (H 5 m, section diameter + 1.1 m,
fck 40 N/mm2, concentrated mass at the top of the
pier 410,000 kg, geometric reinforcement ratio rl 2.41% andhorizontal design load Hd 868 kN, initial cracking moment for
the section Mcr 1.13 MN m).
The symptoms that we monitor over time are the fundamental
period of the piers, whose variation is associated with the
decrease in stiffness, and an equivalent viscous damping. Damp-
ing is obtained from forced vibrations (vibrodyne).
The difference with the model used in Section 5 concerns
essentially in damping: the results obtained on reinforced
concrete structures, in fact, have shown that, in this material,stress intensity, i.e., cracking state, has a decisive influence on
ARTICLE IN PRESS
3000
2500
2000
1500
1000
500
0 10 20 30 40 50
Time of the maintenance intervention (years)
0.2
0.5
L = T /T0
= 0.1
Generalized cost without structure monitoring
Generalized cost with structure monitoring
Totalcost
/m2
Fig. 6. Generalized cost as a function of the time of the maintenance intervention:
curves for different evolutions of the natural period (discount rate n 2%)U
Table 1
Bridge deck: cost parameters associated to the chosen type of intervention [9].
Type of intervention Fiber-reinforced polymer
attaching
Fixed part of the intervention cost, C0 400$/m2
Time period encompassed by the analysis of costs, th 50 years
Cost parameter, s 230
Cost parameter, q 2
The coefficient reflecting the risk cost, cf 4000$/m2
The discount rate of money, v 2%
Parameters associated with parabolic function for
multiplier, l
p1 0.25
l p1b2+p2b+p3 p2 2.0
p3 5
3000
2500
2000
1500
1000
500
0
0
10 20 30 40 50
Time of the maintenance intervention (years)
c =0.35
c =0.35
c
=0.45
c =0.45
c =0.55
c =0.55
1
0.5
0 0.005 0.024 0.043 0.175 0.194 0.213
CDF
To
talcost
/m2
Fig. 7. Generalized cost as a function of the time of the maintenance intervention
(mean value): (a) curves for different values of the variation coefficient ofa, ca and(b) cumulative distribution functions for different values of the variation
coefficient ca.
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equivalent damping. The value of this parameter is seen to
increase with increasing stress level until the structural element is
fully cracked; after cracking, damping begins to decrease [23]. In
the application described below, the evolution of damping for
purposes of reliability assessment is determined with reference to
the conditions that precede the fully cracked state. To this end, a
hysteretic RambergOsgood mechanical model has been adopted
for the pier [24].
The evolution over time of the equivalent stiffness, K, is worked
out from the fundamental period via the elastic damage model
reported above [17], and therefore the RambergOsgood model is
updated on a yearly basis. By exciting the structure by means of a
vibrodyne, for each step of the analysis it is possible to quantify
the corresponding equivalent damping, xeq, according to thefollowing formula [24]:
xeqt 2
p1
2
g 1
1
Fvib=Dvibt
Kt
, (33)
where Fvib is the dynamic load ideally generated by the vibrodyne,
and Dvib is the ensuing displacement observed in the structure.
If the pier is excited yearly with a vibrodyne calibrated at a
constant value (Fvib 300kN), the evolution over time of the
equivalent damping is determined from Eq. (33), in the assump-
tion that the measured horizontal displacement decuples asymp-
totically its initial value and for g 2. As g influences strongly thevariation of damping over time, in practice this parameter should
be determined on a preliminary basis with sufficient accuracy. The
relative variation of damping over time, as plotted in Fig. 8, clearlyshow a potentially increased sensibility of the reliability assess-
ment procedure when also damping is monitored. The reason for
this improvement is that, while a detectable change in the
fundamental period is usually restricted to the first years of
service, the damping parameter continues to increase slowly and
consistently with the bridge effective age.
The reliability of the monitored bridge piers, as a function of
small variations of the logistic vector, L, is worked out from Eq. (12).
In this case, the logistic vector, L, reflects both parameters monitored,
i.e., fundamental period, T, and equivalent damping, xeq;DLTcontains
the deviations of these parameters over time, Tm and xeq,m, from
their primary values, T0 and xeq,0 and Eq. (30) becomes:
RS; LjL0 DL R0S 1 DT
T0;Dx
eqxeq;0
& ' p1p2
!lnR0S
( ), (34)
where qg/qL reduces to a weight vector (p1,p2)T to be associated
with the two symptoms and the accuracy afforded in their
evaluation.
8. Conclusions
This paper addresses the problem of the probabilistic assess-ment of the reliability of civil structures through a symptomatic
approach, which is able to create an appropriate theoretical
framework for taking into account, in safety checks, periodic or
continuous monitoring activities. In particular, it lends itself to the
use of dynamic parameters (frequencies, modal shapes and
damping), identified either through non-destructive tests per-
formed on existing structures or through experimental modal
analyses conducted on structures set up to this end, for the
estimate of the residual lifetime of a construction. By relating
damage to reliability with respect to collapse, the generalized
maintenance cost was also analyzed in order to verify the
economic advantages offered by monitoring.
Simulated applications to bridge structures, subjected to
periodic monitoring, have been illustrated, in which two symp-toms were considered: the reduction in stiffness and the increase
in an equivalent viscous damping. The examples showed that the
outcome of dynamic monitoring systems in bridge structures
might be conditioned by the availability of accurate damping
measurements, which requires ad hoc excitation. While damp-
ing monitoring from forced vibrations may prove very costly, it is
also true that advances in output-only identification techniques
are expected for the next years.
In actual practice measurements are noisy and affected by
different factors, whose relative importance varies with the
structural class and the monitoring system. For instance modal
quantities are known to be strongly affected by thermal fluctua-
tions. A future development of this study will consist of analyzing a
few monitoring systems by expressing measurement uncertainty.
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