Eigenproperties of suspension bridges with damage

Annibale Luigi Materazzia, Filippo Ubertinib

Department of Civil and Environmental Engineering, University of Perugia, Via G. Duranti 93, 06125 Perugia, Italy

amaterazzi@unipg.itbCorresponding author, telephone: +39 075 585 3954; fax: +39 075 585 3897; e-mail: ubertini@unipg.it

Abstract

The vertical vibration of suspension bridges with a damage in the main cables is studied using acontinuum formulation. Starting from a model for damaged suspended cables recently proposedin the literature, an improved expression for the dynamic increment of cable tension is derived.The non-linear equation of motion of the damaged bridge is obtained by extending this modelto include the stiffening girder. The linear undamped modal eigenproperties are then extracted,in closed-form, from the linearized equation of motion, thus generalizing to the presence of anarbitrary damage the expressions known from the literature for undamaged suspension bridges.The linear dynamics of the damaged bridge reveals to be completely described by means of thesame two non-dimensional parameters that govern the linear dynamics of undamaged bridgesand which account for the mechanical characteristics of both the main cable and the girder, withthe addition of three non-dimensional parameters characterizing damage intensity, position andextent. After presenting the mathematical formulation, a parametric analysis is conducted withthe purpose of investigating the sensitivity of natural frequencies and mode shapes to damage,which, in fact, is a crucial point concerning damage detection applications using inverse methods.All through the paper, systematic comparisons with finite element simulations are presented forthe purpose of model validation.

Keywords:suspension bridge, suspended cable, damage detection, linear dynamics, structural healthmonitoring

1. Introduction

As a matter of fact, bridge engineers are called, today, to face a steadily growing demand forlighter and longer bridges, with the consequence that these structures are increasingly exposedto the risk of sustained vibrations which may lead to fatigue problems. These last, combinedwith material aging and chemical-physical interactions with the environment, may determine theonset of damage in structural components, with onerous social and economical costs. Hence,in view of the optimal allocation of public resources for health monitoring and maintenance ofnew structures and for rehabilitation of older ones, a significant increase in interest has arisen,during the last decade, towards simple non-destructive methods aimed at assessing the structuralintegrity of bridge structures.

Among the various approaches for structural health assessment, dynamic methods based onmonitoring the dynamic response of the structure have recently received much attention in thePreprint submitted to Journal of Sound and Vibration February 4, 2014

literature (see for instance [1, 2, 3, 4, 5]) due to the relative simplicity and moderate cost ofdynamic measurements. Some of the available diagnostic methods based on dynamic measure-ments rely on the correlation between damage and modal properties, such as natural frequencies,mode shapes, modal curvatures and so on, which can be estimated from the recorded struc-tural response, by using an appropriate configuration of monitoring sensors [6] and by applyingsuitable system identification methods [7, 8, 9]. Meanwhile, different approaches have been de-veloped in the literature, either requiring [2], or not [10], a reference model for the undamagedstructure.

The response of suspension bridges is usually analyzed using detailed discrete finite elementmodels which allow a proper description of bridges [11] accounting for local details. However,the relatively simple geometry of these structures makes continuum approaches still very attrac-tive, likewise, for instance, in the case of suspended cables [12, 13] and girder bridges [14].Therefore, for the purpose of validating diagnostic models and for preliminary design analysis[15], simple continuum formulations able to describe a general bridge with a minimum numberof non-dimensional parameters are perhaps more appropriate than finite element models. In thisregards, the classic continuum model for the linear vertical vibration of suspension bridges wasfirst proposed by Bleich et al. [16], with successive improvements by Kim [17], and recentlyreviewed by Luco and Turmo [18]. This model considers the linearized equation of motion of anelastic single-span bridge with a shallow parabolic main cable, a dense arrangement of inexten-sible hangers and an Euler-Bernoulli beam to reproduce the girder. It shows that the linear vibra-tion of the considered suspension bridge model is completely governed by two non-dimensionalparameters: the classic Irvine parameter of suspended cables, first introduced by Irvine [19, 20],and a second parameter accounting for the relative stiffness of the girder with respect to the maincable.

Despite the significant interest in suspension bridge modeling, analytical formulations forthese structures which include some kind of damage in their components are quite rare. In thisregard, probably the most meaningful damage condition which should be considered is repre-sented by a sectional loss at some location of the main cables, as the high-strength wires of thesestructural components are known to suffer from corrosion-fatigue problems (see for instanceBetti et al. [21]). The effect of this kind of damage on the static and dynamic response of sus-pended cables has been recently studied by Lepidi et al. [22, 23]. Nonetheless, a similar studyfor suspension bridges is still missing. The model proposed by Lepidi et al. [22] constitutes anextension of the classic Irvine model for suspended cables with an arbitrary sag, showing thatthe damage in the cable determines a sag augmentation and a tension loss. For a fixed damageposition, intensity and extent, these two quantities depend only upon the Irvine parameter of thecable. Consequently, the linear dynamic response of damaged suspended cables is completelydescribed by the Irvine parameter and three non-dimensional damage parameters.

In order to develop appropriate techniques for damage detection in suspension bridges, aresearch program has been started with the following steps: continuum formulation; sensitiv-ity study; environmental effects; analytical wind response modeling; system identification anddamage detection. The main topic of the present paper is to present a continuum model for sus-pension bridges with a shallow parabolic damaged main cable. To this end, the model by Lepidiet al. [22], describing the finite vertical motion of damaged suspended cables, is considered atfirst and the expression of the additional cable tension is improved in such a way to emphasizeits dependence on damage position and to make it consistent with the expression derived byIrvine [19] in the undamaged case. Then, the non-linear equation of motion governing the verti-cal vibration of the damaged bridge is obtained by introducing the flexural term associated with

2

Figure 1: Continuum model of suspension bridge with damaged cable (un-stretched profile dashed lines, static profilecontinuous lines).

the girder. The linearization of this equation around the static reference configuration is solvedin the case of undamped free vibration by deriving the linear modal eigenproperties in closedform, which generalize those known from the literature [18] for undamaged suspension bridgesto the presence of an arbitrary damage. A parametric study is also performed in order to discussthe sensitivity of natural frequencies and mode shapes to damage position, intensity and extentin view of damage detection applications using inverse methods. In this regard, the changes inmodal parameters due to damage are also compared to those produced by temperature variations,either computed via finite element simulations or taken from experimental studies available inthe literature [24, 25].

2. Problem definition

Let us consider the vertical response of a suspension bridge composed of a main cable, astiffening girder (the bridge deck) and a uniform distribution of vertical hangers (see Figure 1).The main cable, hinged at fixed anchors placed at the same vertical elevation, is modeled as amono-dimensional linearly elastic continuum with negligible flexural and shear rigidities. Thedeck is modeled as a uniform, linearly elastic beam, simply supported at its ends. Shear and axialdeformations of the deck are ignored. The hangers are considered to be uniformly distributed,massless and inextensible. The total span of the main cable and the girder, that is, the distancebetween the supports, is equal to L. The Young moduli of the materials constituting the cableand the deck are denoted as Ec and Ed, respectively, while Id is the inertial momentum of thestiffening girder.

Let the curvilinear abscissa, s, be defined along the un-stretched cable profile, whose totallength is equal to L0. In the static reference configuration, the cartesian coordinates of the cableprofile are x(s) and y(s), and the stretched curvilinear abscissa is equal to p. The initial profile ofthe deck is chosen such that, after the application of the dead load, it becomes perfectly straightand horizontal.

A damaged region in the suspension cable is considered. Namely, following relevant litera-ture works [1, 22], the cable portion placed between a1 < s < a2 is characterized by a reduced

3

cross-section equal to Ac(1 − η), η being a damage intensity factor, with 0 ≤ η < 1, and Ac beingthe undamaged cross-section.

In order to derive the equation of motion of the considered mechanical model in closed form,it is necessary to compute the exact deformed configuration of the damaged bridge under self-weight (x(s), y(s)), by solving the geometrically nonlinear elastic equilibrium equations. In lightof the analytical complexity of this task, an exact modeling of this problem would probably pre-clude the possibility of deriving a closed form solution. However, appropriate assumptions allowto simplify this task essentially without any significant loss of accuracy.

First of all, considering that, in real suspension bridges, the flexural stiffness of the deck ismuch smaller than the geometric stiffness of the cable, the influence of the deck on the static re-sponse can be disregarded with a good approximation. Furthermore, by invoking the hypothesisof a moderate sag-to-span ratio, which is again quite reasonable considering that this ratio in realsuspension bridges is usually equal to about 1/10, it is generally acceptable [18] to assume thedeck weight to be uniformly distributed along the curvilinear cable profile. Thus, the deformedconfiguration of the bridge can be approximated with that of the damaged main cable subjectedto its self-weight and to the dead load of the deck, with the vertical deflection of the deck beingequal to the vertical deflection of the cable.

The exact deformed configuration under self weight of a damaged suspended cable withan arbitrary sag was derived in closed form by Lepidi et al. [22]. Here, this solution is alsoadopted for the problem at hand considering a heavier cable whose dead load per unit-lengthis the combination of cable and deck weights. The static solution of the bridge obtained underthese hypotheses is reviewed in the Appendix. Its accuracy will be checked by independent finiteelement simulations in the successive development of this work.

As in the work by Lepidi et al. [22], the effect of damage on the static configuration of thebridge is characterized by means of the following two factors:

κ2 =ffu

(1)

χ2 =HHu

(2)

whose calculation follows from the computation reported in the Appendix. In Eqs. (1-2) f and fuare the sag of the cable in the damaged and undamaged cases, respectively, while H and Hu arethe corresponding horizontal components of cable tension at the anchors. The vertical reactionat the boundary is denoted as V , while the cable tension is denoted as T .

It is known [22] that factors κ2 and χ2 in Eqs. (1-2) are functions of the non-dimensionalIrvine parameter [19], λ2, of the main cable, which is defined in the successive developmentof this work, the damage parameter η and two other damage parameters, δ and γ, which definedamage extension and position. These two parameters are chosen as:

δ =x2 − x1

L(3)

γ =x1 + x2

2L(4)

where, for convenience, the damaged region, defined in terms of the x coordinate, is x1 < x < x2,where x1 = x(a1) and x2 = x(a2). It is also worth recalling that, whilst the sag augmentationfactor is directly affected by η, δ and γ, the tension loss factor χ2 is directly affected by η and δ,while it is essentially unaffected by γ. This means that a change in damage position is expectednot to produce significant variations of χ2.

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Figure 2: Bridge static (dashed lines) and dynamic varied (continuous lines) configurations (the damaged portion of thecable is highlighted).

3. In-plane vibration of a suspension bridge with damaged cable

3.1. Equation of motionUnder the assumption of a moderate sag-to-span ratio, the stretched curvilinear abscissa p

can be approximated with the un-stretched curvilinear abscissa s, the x direction coincides withthe direction of the deck in the equilibrium configuration and the reference static configurationof the cable can be described by the parabola y(x) given by:

y(x) = 4κ2 fux(L − x)

L2 (5)

It is worth noting that the parabolic assumption is known to be very accurate, even in the dam-aged case, for usual values of the sag-to-span ratio fu/L [22].

Assuming the static configuration of the bridge as the reference one, the dynamic variedconfiguration is described by the displacement functions u(x, t) and v(x, t) in the x and y direc-tions, respectively (see Figure 2). Due to the assumption of a dense arrangement of inextensiblehangers, v(x, t) denotes the vertical deflection of both the deck and the main cable, while u(x, t)represents the horizontal displacement of the main cable. The horizontal displacement of thedeck is herein disregarded.

Since the cable profile is shallow, it is acceptable to invoke the classic hypothesis of negli-gible inertial forces in the longitudinal direction [18, 19, 22] and to disregard the longitudinalcomponent u. In this case, the horizontal component of the increment of cable tension, ∆h(t), isspatially uniform, as no longitudinal forces are acting. By introducing the linear elastic constitu-tive equation for the cable, ∆h(t) can be written as follows:

∆h(t)dsdx= EcAc(1 − ζ(x))ϵ(x, t) (6)

In Eq. (6), ϵ(x, t) = dsI−dsds is the Lagrangian increment of the strain in the cable, where ds =√

dx2 + dy2 denotes the length of the infinitesimal cable element in the reference static config-uration and dsI =

√(dx + du)2 + (dy + du)2 the corresponding length in the dynamic varied

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configuration, while the function ζ(x) is defined as:

ζ(x) =

η for x1 < x < x2

0 elsewhere(7)

Equilibrium conditions at the boundaries of the damaged region also require that:

EcAcϵ(x1, t)− = EcAc(1 − η)ϵ(x1, t)+ EcAc(1 − η)ϵ(x2, t)− = EcAcϵ(x2, t)+ ∀t (8)

where the superscripts + and − denote right and left limits, respectively.By integrating Eq. (6) in space and retaining terms up to the second order, the following

equation is thus obtained:∫ L

0

∆h(t)EcAc(1 − ζ(x))

(dsdx

)3dx =

∫ L

0

(u′ + y′v′ +

12

(v′)2)dx (9)

where a prime denotes the derivative with respect to x. Making use of Eqs. (5) and (8) andintegrating by parts the right hand side of Eq. (9) using the boundary condition u(0, t) = u(L, t) =0, ∀t, the following equation is thus obtained from Eq. (9):

∆h(t) =EcAc

LueΓ

∫ L

0

(y′v′ +

12

(v′)2)dx (10)

where the factor Γ is obtained as:

Γ =1 − η(1 − δ L

Le

)+

8κ4 f 2u ηδ

LLe

(3 − 12γ + δ2 + 12γ2)

(1 − η)Le

Lue

(11)

using the following approximation:(dsdx

)3�(1 +

y′(x)2

2

)3�(1 +

32

y′(x)2)

(12)

In Eqs. (10,11) Le =∫ L

0 ( dsdx )3dx � L(1 + 8(κ2 fu/L)2) is the effective length of the cable [20]

in the damaged configuration, while Lue � L(1 + 8( fu/L)2) is the same quantity evaluated in the

undamaged configuration. The factor Γ defined in Eq. (11) depends upon the sag-to-span ratiofu/L, the damage intensity factor η, the damage extension factor δ and the damage position factorγ. It can be noted that the asymptotic value of the expression of ∆h(t) in Eq. (10) for small valuesof fu/L is equal to:

lim fu/L→0∆h(t) =EcAc

L(1 − η)

1 − η(1 − δ)∫ L

0

(y′v′ +

12

(v′)2)dx (13)

which coincides with the expression reported in [22] where the approximation (ds/dx)3 � 1 wasused, coherently with the assumption of fu/L small. As a consequence of assuming (ds/dx)3 � 1,the dependence on the damage position parameter γ is not present in Eq. (13). On the contrary, byretaining the term (ds/dx)3 in Eq. (9), the factor γ appears in Eq. (11) thus properly consideringthe role played by damage position on the additional cable tension. Moreover, the same term is

6

responsible for the appearance, in Eq. (11), of the effective length of the cable (which, contrarilyto what was done in Eq. (13), is not approximated with L), consistently with Irvine’s theory [19].

Under the assumed hypotheses, likewise in the case of undamaged suspension bridges [18],the equation governing the finite motion of the bridge with a shallow damaged main cable cannow be expressed as that of a damaged suspended cable [22], with the addition of the flexuralstiffness term associated with the girder and considering the combined mass per unit length, m,of the deck and the cable in the calculation of the vertical inertial force:

mv̈ + EdIdv′′′′ − χ2Huv′′ − (y′′ + v′′)∆h(t) = 0 (14)

where the horizontal component of the static cable tension is equal to χ2Hu, its dynamic incre-ment ∆h(t) is given by Eq. (10) and a dot denotes the derivative with respect to time t.

The solution of Eq. (14) must satisfy the following boundary conditions:

v(0, t) = v′′(0, t) = 0 v(L, t) = v′′(L, t) = 0 ∀t (15)

alongside the following compatibility conditions:

v(x1, t)− = v(x1, t)+ v(x2, t)− = v(x2, t)+ ∀tv(x1, t)′− = v(x1, t)′+ v(x2, t)′− = v(x2, t)′+ ∀tv(x1, t)′′− = v(x1, t)′′+ v(x2, t)′′− = v(x2, t)′′+ ∀tv(x1, t)′′′− = v(x1, t)′′′+ v(x2, t)′′′− = v(x2, t)′′′+ ∀t

(16)

3.2. Linear normal eigenpropertiesWith the purpose of deriving the linear eigenproperties of the bridge, Eq. (14) must be

linearized around the static configuration which, making use of Eq. (5) and integrating by partsthe right hand side of Eq. (10) using the boundary conditions in Eq. (15), yields:

mv̈ + EdIdv′′′′ − χ2Huv′′ +EcAc

LueΓ

(8κ2 fuL2

)2 ∫ L

0vdx = 0 (17)

Eq. (17) is a linear integral-differential equation which coincides with the classic one [18] in theundamaged case (i.e. when η = 0 and, consequently, χ2 = 1) and with the Irvine model [19] oflinear cable vibration when Id = 0 and m is interpreted as the mass per unit length of the cable.

By introducing the non-dimensional time t = t√m/HuL

and the non-dimensional variablesx = x/L, v = v

8 fu, Eq. (17) can be rewritten in the following form:

v̈ + µ2v′′′′ − χ2v′′ + κ4λ2

Γ

∫ 10 vdx=0 (18)

The Irvine parameter λ2 appearing in Eq. (18) is the one that governs the linear theory of vibra-tion of suspended cables [19] while µ2 is another non-dimensional parameter which quantifiesthe relative significance of the bending stiffness of the deck compared to the geometric stiffnessof the cable [18]. These two parameters are defined as follows:

λ2 =

(8 fuL

)2 LLu

e

EcAc

Hu(19)

µ2 =EdId

L2Hu(20)

7

The boundary conditions of Eq. (18) are readily obtained from Eq. (15) as:

v(0, t) = v′′(0, t) = v(1, t) = v′′(1, t) = 0 ∀t (21)

In the case of free vibration, a solution of Eq. (18) with the following form is desired:

v = ϕ(x)eiωt (22)

i being the imaginary unit and ω denoting the circular frequency. Substituting Eq. (22) in Eq.(18) yields the following condition:

µ2ϕ′′′′ − χ2ϕ′′ − ω2ϕ = −κ4λ2

Γ

∫ 1

0ϕ(x)dx (23)

where ω = ωL√

m/Hu. Eq. (23) can be rewritten in the following compact form:

µ2ϕ′′′′ − χ2ϕ′′ − ω2ϕ = − κ4λ2

Γh

h =∫ 1

0 ϕ(x)dx(24)

In the case of antisymmetric vibration, the integral∫ 1

0 ϕ(x)dx is identically zero, i.e. thedynamic increment of cable tension is zero, and the mode shapes and the corresponding naturalcircular frequencies are readily obtained from Eq. (24) as:

ϕ(x) = Cnsin(2nπx)

ω = 2πnχ√

1 +(2πn µ

χ

)2 (25)

with Cn representing a set of amplitude coefficients.In the case of symmetric vibration the mode shapes satisfy:

ϕ(x) = h κ4

Γ

( λω

)2[1 − ( β2

β2+α2 ) cos(α(x−1/2))cos(α/2) − ( α2

β2+α2 ) cosh(β(x−1/2)cosh(β/2)

](26)

with α2 and β2 being defined as:

α2 =χ2

(√1+(2ω µχ2

)2−1

)2µ2

β2 =χ2

(√1+(2ω µχ2

)2+1

)2µ2

(27)

Substituting Eq. (26) in the expression of h appearing in Eq. (24) the following characteristicequation is obtained:(

ω

λ

)2=κ4

Γ

(1 − β2

α2 + β2

tan(α/2)α/2

− α2

α2 + β2

tanh(β/2)β/2

)(28)

which can be solved in the unknown ω.It is important to note that the expressions reported in Eqs. (25-28) correspond to those of

an undamaged bridge, known from the literature [16], when η = 0. Also, the same expressions8

correspond to those of a damaged suspended cable reported in reference [22] when µ2 = 0 and Γis deduced from Eq. (13), and to the classic Irvine solution of the linear cables vibration whenη = 0 and µ2 = 0.

Looking at the previous derivations, the problem of calculating the linear vertical eigen-properties of suspension bridges with a damaged main cable is solved under the assumptions ofa moderate sag-to-span ratio, a parabolic profile for the damaged main cable and a negligibleeffect of the bending stiffness of the girder on the static response of the bridge. The derivedequations will be adopted in the next section to investigate the sensitivity of modal properties todamage in view of the application of damage detection techniques based on vibration monitor-ing. Also, the accuracy of the derived expressions will be checked by means of independent finiteelement simulations, using realistic values of λ2 and µ2 and without invoking the aforementionedsimplifications adopted in the analytical model.

3.3. DiscussionBefore commenting the parametric study, a few observations on the derived expressions

should be made. First of all, considering Eq.(25) it can be observed that the mode shapes ofantisymmetric modes are damage-independent. Furthermore, the ratio between the circular fre-quencies of antisymmetric modes in the damaged, ω, and undamaged, ωu, cases is equal to:

ω

ωu

∣∣∣∣∣asymm

= χ

√√1 + (2πnµ)2

χ2

1 + (2πnµ)2 (29)

which only depends upon µ2 and the tension loss factor χ2. This means that the damaged nat-ural frequencies ω of antisymmetric modes directly depend upon λ2, µ2, η and δ, whilst smallvariations with γ are expected since χ2 is only slightly affected by such a factor. Therefore,the identification of antisymmetric modes can laboriously allow to localize the position of thedamage while permitting, in principle, to identify damage intensity and extent. Moreover, for anincreasing modal order n and for large values of µ2, Eq.(29) yields:

limn→∞ω

ωu

∣∣∣∣∣asymm

= limµ2→∞ω

ω u

∣∣∣∣∣asymm

= 1 (30)

which means that the sensitivity of natural frequencies of antisymmetric modes to damage de-creases with increasing modal order and bending stiffness of the girder. The maximum sensitivityis thus obtained for n = 1 and µ2 = 0 which gives ω

ωu|asymm = χ and which coincides with the

result obtained in [22] for suspended cables.Similar observations for symmetric modes are not as straightforward. Indeed, Eqs. (26-28)

show that not only the natural frequencies, but also the mode shapes, of symmetric modes are af-fected by damage. Moreover the effect of damage changes with the modal order in a way whichis difficult to predict due to the significant nonlinearity of the characteristic equation, whosesolutions vary with the damage parameters in a relatively complex manner. In particular, thesymmetric mode that is expected to exhibit the maximum sensitivity with respect to damage isnot necessarily the lowest one but the one that, for the fixed value of µ2, is the most sensitive tovariations of the Irvine parameter λ2 in the linear spectrum. Indeed, damage produces a changein the term κ4

Γwhich, in Eq. (28), acts equivalently to a change of the Irvine parameter.

It is also important to mention that either the natural frequencies and the mode shapes ofsymmetric modes depend upon λ2, µ2, η, δ and γ. Particularly, the dependence on the damage

9

parameter γ, which is related to its effect on κ2 and to its appearance in Eq.(11), ensures, in prin-ciple, the possibility of identifying not only damage intensity and extent, but also its location,using the information on the identified symmetric modes.

4. Parametric study

4.1. Effects of damage on the static response

First of all, the effects of damage on the static response of the bridge are worth evaluatingconsidering realistic values of the non-dimensional parameters λ2 and µ2 which fully character-ize the behavior of undamaged suspension bridges. In this regard, Luco and Turmo [18] reportedthat, for a representative sample of 14 real suspension bridges in the world, the cable parameterλ2 ranges from 148 to 376 with an average of 220, while the relative girder stiffness parameter µ2

varies approximately in the range 0.5 ·10−3 to 10.0 ·10−3 with an average of 1.17 ·10−3. Here, thevalues λ2 = 240 and µ2 = 3.5 · 10−3 are first considered, as they characterize the New CarquinezBridge, which is a suspension bridge located in California, USA, that was studied in a recentpaper by one of the Authors [11]. The sag-to-span ratio of such a bridge is about equal to 0.10.Two more cases, resulting either from an increase or a reduction of the deck mass and an increaseof the inertial momentum of the deck of the NCB, are also considered: λ2 = 120, µ2 = 1.5 · 10−3

and λ2 = 360, µ2 = 10 · 10−3.

Figure 3: Static response of suspension bridges with damage (in the cable plot the damaged portion of the cable ishighlighted). (a) sag augmentation factor κ2 versus damage intensity η (γ = 0.5, δ = 0.2); (b) κ2 versus η (γ = 0.25,δ = 0.2); (c) tension loss factor χ2 versus η (γ = 0.5, δ = 0.2); (d) χ2 versus η (γ = 0.25, δ = 0.2).

10

The results of a refined finite element model (FEM) simulation are also presented. TheFEM model is used both for calculating the non-linear static response and for performing thesubsequent modal analysis. It is a numerical two-dimensional model, representing, as much aspossible, the continuum mechanical model shown in Figure 1. Two-nodes finite strain beam el-ements are used to model the deck and the main cable, considering, in this last case, near zerobending stiffness. A consistent formulation is adopted for modeling the mass of the main ca-ble, while the mass of the deck is lumped at the deck nodes. Two nodes linear elastic beamelements are used to model the massless hangers having near-zero bending rigidity and ”large”axial stiffness, obtained by means of an elastic material having a Young modulus three orders ofmagnitude greater than the one of the material constituting the deck. The self-weight is derivedfrom the elements’ masses. The solution procedure is composed by a large displacement incre-mental non-linear static analysis under self-weight and a subsequent modal analysis consideringthe last solution obtained at the end of the static step. The spatial discretization, chosen after apreliminary sensitivity analysis, consists of 500 finite elements both for modeling the deck andthe main cable. The non-linear static solution is achieved by means of about 3000 incrementalsteps, calculating the equilibrium at each step by means of a modified Newton-Raphson proce-dure. Using a standard machine, for a specific damage condition, the non-linear static solutionrequired about 20 minutes to be solved. The computational cost of the modal analysis was, onthe contrary, almost negligible.

It should be noticed that the hypotheses assumed in the analytic model, that is, neglecting theflexural stiffness of the deck and distributing the weight of the deck along the curvilinear abscissaof the cable in calculating the static response, as well as the parabolic approximation made inthe derivation of the equation of motion, are not invoked in the FEM simulation. Thus, the FEMmodel is adopted for the purpose of validating the analytic one.

In order to appreciate significant effects of the damage on the static response of the bridge, adamage extension parameter δ = 0.2 is here considered, while two cases of damage position areinvestigated: damage at mid-span (γ = 0.5) and damage at quarter-span (γ = 0.25). The damageintensity factor η is varied between η = 0 (undamaged case) and η = 0.5.

Figure 3 shows the variation of the sag augmentation factor κ2 and the tension loss factor χ2

with η in all the considered cases, including the results of the FEM simulation. The first observa-tion is that the differences between analytic results and finite element predictions are practicallynegligible, which confirms the adequacy of the hypotheses made in the analytic model for calcu-lating the static response.

Looking at Figure 3 it is also worth noting that the effects of damage on the static responseare very small (sag augmentations and tension losses are of the order of a few percent), evenfor large values of η. This circumstance seems to preclude any possibility of detecting damageby measuring the vertical deflection of the cable and/or its strain. Moreover, a variation in theposition of the damage is seen to have a very small effect on the sag augmentation factor κ2 whilenot affecting at all the tension loss factor χ2, likewise in damaged suspended cables [22].

4.2. Effects of damage on the dynamic responseThe effects of damage on the natural frequencies and mode shapes of the bridge are here

evaluated in order to discuss, to some extent, the applicability of damage detection techniquesbased on vibration monitoring for the problem under investigation. The same cases analyzed inSection 4.1 are considered.

Figures 4 and 5 show the ratios between damaged and undamaged natural circular frequen-cies (ω and ωu respectively) of the first 8 modes of the structure (4 antisymmetric and 4 symmet-

11

Figure 4: Sensitivity of natural frequencies to damage at mid-span (in the mode shape plot the damaged portion of thecable is highlighted). Mode no. (A: antisymmetric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g) 4A;(h) 4S.

ric modes) for damage placed at mid-span and quarter-span, respectively. Higher order modesare not considered here, because, in practical applications, their identification, although possiblewith sophisticated techniques, is more problematic than identifying lower order modes [11]. Thevalues of the modal assurance criterion (MAC) between damaged and undamaged modes forvarying η, and for damage at mid-span and quarter span, are also shown in Figures 6 and 7. FEMresults, using the model presented in Section 4.1, are also shown in Figures 4,5,6,7.

The results confirm that the mode shapes of antisymmetric modes are not affected by damageand that the sensitivity of natural frequencies of antisymmetric modes to damage decreases asthe order of the modes increases, according to Eq. (29). Moreover, the results also confirm thatthere is not a trend of the sensitivities of natural frequencies of symmetric modes with the modalorder. Indeed, for the values of the mechanical parameters under consideration, the second sym-metric mode is the one that exhibits the largest sensitivity to damage, both in terms of naturalfrequency and mode shape. The first symmetric mode is also appreciably affected by damage,while higher order symmetric modes are seen to be relatively unaffected. It is also worthwhileto note that damage position has a very slight influence on modal parameters which clearly indi-cates a technical difficulty in locating damage using modal information. Also, considering theirlager sensitivity with respect to damage, the use of natural frequencies for detecting damage inthe main cables of suspension bridges is certainly more promising than that of mode shapes. Onthis respect, it should be also mentioned that small variations of natural frequencies are much

12

Figure 5: Sensitivity of natural frequencies to damage at quarter-span (in the mode shape plot the damaged portion ofthe cable is highlighted). Mode no. (A: antisymmetric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g)4A; (h) 4S.

13

Figure 6: Sensitivity of mode shapes to damage at mid-span (in the mode shape plot the damaged portion of the cable ishighlighted). Mode no. (A: antisymmetric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g) 4A; (h) 4S.

14

Figure 7: Sensitivity of mode shapes to damage at quarter-span (in the mode shape plot the damaged portion of the cableis highlighted). Mode no. (A: antisymmetric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g) 4A; (h)4S.

15

Figure 8: Sensitivity of natural frequencies to damage intensity and extent (λ2 = 240, γ = 0.5). Mode no. (A: antisym-metric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g) 4A; (h) 4S.

16

Figure 9: Effects of temperature variation on natural frequencies via finite element simulations (ωT denotes the circularfrequency as a function of the temperature variation ∆T , while ω is the circular frequency for ∆T = 0). Mode no. (A:antisymmetric, S: symmetric): (a) 1A; (b) 1S; (c) 2A; (d) 2S; (e) 3A; (f) 3S; (g) 4A; (h) 4S.

easier to be experimentally observed than small variations of mode shapes. Indeed, if the cor-relation between damaged and undamaged mode shapes, quantified in terms of MAC values, isto be used for damage detection, a large number of sensors are required to reliably compute theMAC. On the contrary, the accurate identification of natural frequencies does not generally entailsuch a need for a dense arrangement of monitoring sensors.

In order to emphasize the role played by damage extension on the variations of natural fre-quencies, Figure 8 shows the ratios between damaged and undamaged frequencies for varying ηand δ in the case λ2 = 240 and γ = 0.5. The results show that, as expected, as the damage extentis reduced, the variations of natural frequencies also decrease.

With the purpose of interpreting the obtained results, it should be observed that the key aspectto be addressed in damage detection using dynamic methods is the capability to identify, localizeand quantify structural damage especially at an early stage, i.e. when it cannot be detected by amere visual inspection and there is no clue of a possible structural deficiency. Broadly speaking,the minimum level of detectable damage is the one that determines the minimum variation ofthe considered damage index which is not hidden by environmental effects nor by random errorsin system identification. In this respect, looking at Figure 8 it can be observed, for instance,that a relatively small damage in the main cable consisting of a sectional loss of 15% with anextension of 5% of the total bridge span determines a variation of the natural frequency of the

17

most sensitive mode which is around 0.5%. A finite element analysis has been performed forcomparing this frequency variation with those produced by uniform temperature changes in themain cable. The results are presented in Figure 9 and show that variations similar to those pro-duced by small damages could be expected in some modes, for a temperature change of about50 ◦C, although the modes which exhibit the largest sensitivities to damage are not necessarilythose that exhibit the largest sensitivities to temperature variation. It should be also mentionedthat temperature changes may cause increments or decrements of natural frequencies which mayeither hide or enhance the effects of damage. Even more significant temperature effects wereobserved in the literature in real suspension bridges. Yang et al. [24] reported, for instance,that frequency variations larger than 1%, and produced by temperature changes of about 40 ◦C,where experimentally observed in a long-span suspension bridge, the Runyang Bridge, China. Itis apparent, therefore, that the detection of small damages in suspension bridges via vibrationalmeasurements might be possible only by means of a permanent monitoring which allows a clearelimination of environmental factors.

5. Conclusions

An analytical model describing the finite vertical vibration of suspension bridges with dam-age in the main cables has been presented. Starting from an existing non-linear model for dam-aged suspended cables, an improved expression for the additional dynamic cable tension hasbeen derived, emphasizing its dependence on damage position. Then, the non-linear equation ofmotion of the damaged bridge has been obtained by introducing the flexural term associated withthe girder. The linearization of this equation around the static reference configuration has beensolved in the case of free motion by deriving eigenfrequencies and mode shapes of the damagedbridge in closed form. These expressions constitute a generalization to the presence of an arbi-trary damage of those known from the literature for undamaged bridges. Analytical predictionshave been also systematically compared with finite element simulations for the purpose of modelvalidation. The following are the main conclusions of this work:

• The linear vertical dynamics of the damaged bridge is completely governed by the sametwo non-dimensional parameters that govern the linear dynamics of undamaged bridgeswith the addition of three non-dimensional parameters characterizing damage intensity,position and extent.

• The natural frequencies of symmetric modes are generally more significantly affected bydamage than those of antisymmetric ones. Likewise in the case of suspended cables, thiscircumstance is due to the combined effect of tension loss and sag augmentation in thecase of symmetric modes. Instead, only tension loss, and not sag augmentation, affects thefrequencies of antisymmetric modes.

• The sensitivity of natural frequencies of antisymmetric modes with respect to damage de-creases with increasing modal order, while the same sensitivity in the case of symmetricmodes is maximum for some specific mode depending on the relevant mechanical param-eters of the bridge.

• The mode shapes of antisymmetric modes are not affected by damage while those of sym-metric modes are, but very slightly. Hence, the use of frequency variations for damagedetection appears to be, in the case of suspension bridges, comparatively more promisingthan that of mode shape variations.

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• The variations of the vertical modal parameters with damage are small, in the sense that forsmall damages such variations are comparable to those produced by environmental effectssuch as temperature variations. On this respect, a future study devoted to the analyticalmodeling of the torsional response of damaged suspension bridges will be particularlyimportant, as torsional modes might be also sensitive to damage in the main cables andmight probably allow to discern which of the main cables is damaged.

Appendix A.

Under the assumption of a negligible effect of the stiffening girder on the static response, thecalculation of the equilibrium configuration of the bridge essentially follows that presented byLepidi et al. [22] in the case of a damaged suspended cable. For the sake of completeness, thiscalculation is also recalled here.

Given the definitions reported in Section 2, the condition of in-plane cable static deflectionyields [20]:

dp =√

dx2 + dy2 (A.1)

Moreover, the following equilibrium conditions can be written:

Tdxdp= H (A.2)

Tdydp= V − (W + qL)

sL0

(A.3)

where W is the total weight of the cable, while q is the dead load of the deck per unit length. Itshould be noted that, in Eq.(A.3), the dead load of the deck has been uniformly distributed alongthe cable profile: this assumption is acceptable under the hypothesis of a moderate sag-to-spanratio. Substituting Eqs. (A.2) and (A.3) in Eq. (A.1) yields:

T (s) =

√H2 +

(V − (W + qL)

sL0

)2(A.4)

The linear elastic constitutive equation gives:

TEcAc(1 − ζ(s)

) = dpds− 1 (A.5)

where ζ(s) = η for a1 < s < a2 and ζ(s) = 0 elsewhere. Now, making use of Eqs. (A.2), (A.3)and (A.5) the following equations are obtained:

dxds=

dxdp

dpds=

HEcAc(1 − ζ(s))

+H

T (s)(A.6)

dyds=

dydp

dpds=

( VW + qL

− sL0

)( W + qLEcAc(1 − ζ(s))

+W + qL

T (s)

)(A.7)

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which can be solved in the unknown functions x(s) and y(s) satisfying the following boundaryand continuity conditions:

x(0) = 0 x(L0) = 0 y(0) = 0 y(L0) = 0x−1 = x+1 y−1 = y+1 x−2 = x+2 y−1 = y+1

(A.8)

Integration of Eqs. (A.6) and (A.7) with the result obtained in Eq. (A.4) and imposing theboundary conditions for s = 0 in Eq. (A.8) yields:

x(s) =

Hs

EcAc+

HL0W+qLΨ1(s) f or 0 ≤ s ≤ a1

11−η

H(s−ηa1)EcAc

+HL0

W+qLΨ1(s) f or a1 < s < a2

HEcAc

(s + η

1−η (a2 − a1))+

HL0W+qLΨ1(s) f or a2 ≤ s ≤ L0

(A.9)

y(s) =

(W+qL)sEcAc

(V

W+qL −s

2L0

)+

HL0W+qLΨ2(s) f or 0 ≤ s ≤ a1

11−η

(W+qL)EcAc

(s( V

W+qL −s

2L0

)−ηa1( V

W+qL −a1

2L0

))+

HL0W+qLΨ2(s) f or a1 < s < a2

(W+qL)EcAc

(s( V

W+qL −s

2L0

)+

+η

1−η (a2 − a1)( V

W+qL −a1+a22L0

))+

HL0W+qLΨ2(s) f or a2 ≤ s ≤ L0

(A.10)

where:Ψ1(s) = asinh

(VH

)− asinh

(V − (W + qL)s/L0

H

)(A.11)

Ψ2(s) =

√1 +(V

H

)2−

√1 +(V − (W + qL)s/L0

H

)2(A.12)

Imposing in Eqs. (A.9,A.10) the boundary conditions for s = L0 in Eq. (A.8) yields:

HEcAc

(L0 +

η1−η (a2 − a1)

)+

HL0W+qLΨ1(L0) = 0

(W+qL)EcAc

(L0( V

W+qL −12 )+

η1−η (a2 − a1)

( VW+qL −

a1+a22L0

))+

HL0W+qLΨ2(L0) = 0

(A.13)

which can be solved in the unknown reactions H and V .Now, calculating the static solution in both the undamaged (η = 0) and damaged (η > 0)

cases using Eqs. (A.9-A.13), allows to obtain the tension loss factor χ2 and the sag augmentationfactor κ2 defined in Eqs. (2-1).

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