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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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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
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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.

R. Ceravolo et al. / Reliability Engineering and System Safety 94 (2009) 133113391334


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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


Monitored symptom profiles L >0

Monitored symptom profiles L


7/9

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

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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.



8/9

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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1.4

1.3

1.2

1.1

1

eq

/

eqt

=0

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Time (years)

eq

eq, 0=0.3


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9/9

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