Effect of cumulative seismic damage and corrosion on the life-cycle cost of reinforced concrete bridges

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2009; 38:887–905Published online 17 November 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.873

Effect of cumulative seismic damage and corrosion onthe life-cycle cost of reinforced concrete bridges

R. Kumar1, P. Gardoni1,∗,† and M. Sanchez-Silva2

1Zachry Department of Civil Engineering, Texas A&M University, College Station, TX, U.S.A.2Departamento de Ingenierı́a Civil y Ambiental, Universidad de Los Andes, Bogota, Columbia

SUMMARY

Bridge design should take into account not only safety and functionality, but also the cost effectivenessof investments throughout a bridge life-cycle. This paper presents a probabilistic approach to computethe life-cycle cost (LCC) of corroding reinforced concrete (RC) bridges in earthquake-prone regions. Theapproach is developed by combining cumulative seismic damage and damage associated with corrosiondue to environmental conditions. Cumulative seismic damage is obtained from a low-cycle fatigue analysis.Chloride-induced corrosion of steel reinforcement is computed based on Fick’s second law of diffusion.The proposed methodology accounts for the uncertainties in the ground motion parameters, the distancefrom the source, the seismic demand on the bridge, and the corrosion initiation time. The statistics of theaccumulated damage and the cost of repairs throughout the bridge life-cycle are obtained by Monte-Carlosimulation. As an illustration of the proposed approach, the effects of design parameters on the LCC ofan example RC bridge are studied. The results are valuable in better estimating the condition of existingbridges and, therefore, can help to schedule inspection and maintenance programs. In addition, by takinginto consideration the two deterioration processes over a bridge life-cycle, it is possible to estimate theoptimal design parameters by minimizing, for example, the expected cost throughout the life of thestructure. A comparison between the effects of the two deterioration processes shows that, in seismicregions, the cumulative seismic damage affects the reliability of bridges over time more than the corrosioneven for corrosive environments. Copyright q 2008 John Wiley & Sons, Ltd.

Received 20 January 2008; Revised 6 October 2008; Accepted 6 October 2008

KEY WORDS: life-cycle cost; cumulative seismic damage; low-cycle fatigue; reinforced concrete bridges;Monte-Carlo simulation; corrosion

1. INTRODUCTION

Bridges experience various damages and deteriorations during their service life. Therefore, theyneed regular inspections, maintenance, and repairs to ensure that they perform above a minimum

∗Correspondence to: P. Gardoni, Zachry Department of Civil Engineering, Texas A&M University, College Station,TX, U.S.A.

†E-mail: [email protected]

Copyright q 2008 John Wiley & Sons, Ltd.

888 R. KUMAR, P. GARDONI AND M. SANCHEZ-SILVA

performance level at all times. A large amount of funds are required today for the repairs andupgrade of deficient bridges. For example, according to USA TODAY [1], the Federal HighwayAdministration puts the current cost of upgrading bridges at $63 billion. This situation makesoptimum fund allocation and life-cycle cost (LCC) analysis a priority for bridge managementsystems and resource allocation.

The significance of LCC analysis and its application to deteriorating civil infrastructure systemsis illustrated in [2, 3]. Chang and Shinozuka [4] develop a framework for LCC analysis for seismicrisk. Stewart and Val [5] and Li [6] consider the problem of structural deterioration due to corrosionof steel reinforcement. Mori and Ellingwood [7, 8] and Kong and Frangopol [9] discuss the roleof inspections and maintenances in optimizing the performance and cost of bridges.

For civil infrastructure systems, corrosion is primarily considered as the deteriorating agent thataccelerates the vulnerability of bridges to natural hazards such as earthquakes. However, it is wellknown that smaller earthquakes not causing complete failure of a bridge can also cause structuraldeterioration over a period of time due to accumulation of seismic damage. The objective of thispaper is to present a methodology to include the combined effects of cumulative seismic damageand corrosion of reinforced concrete (RC) bridges in the LCC analysis. A variation of the low-cyclefatigue theory [10] that takes into consideration the deterioration in concrete and steel is developedin this paper to model the cumulative seismic damage. To account for the effects of corrosion, theproposed model uses a probabilistic seismic demand model for corroded RC bridges developedby Choe et al. [11]. While the methodology is developed for a bridge with a single column bent,the formulation can be extended to model the cumulative seismic damage in bridges with multiplecolumns. As a practical illustration, the proposed formulation is used to assess the LCC of anexample bridge.

This paper is divided into five sections. The first section presents how the proposed approachaccounts for the rate of occurrence of earthquakes and their magnitudes, and the structural demandparameters. The second section discusses how the failure probability of RC bridges is computedaccounting for the cumulative seismic damage. In the third section, the proposed approach isextended to take into account corrosion. The fourth section of the paper presents the method-ology to compute the LCC of a bridge. Finally, the fifth section presents the conclusions of thepaper.

2. SEISMICITY AND STRUCTURAL DEMAND

The LCC analysis requires first to estimate the seismic characteristics of a region (e.g. earthquakesources and rates of occurrence). This section presents the probabilistic model used in the proposedmethodology to simulate the occurrence and the magnitude of earthquakes. In addition, this sectiondescribes the computation of structural demand parameters such as drift, seismic energy, andnumber of inelastic cycles of the response of an equivalent single-degree-of-freedom (SDOF)system.

2.1. Seismicity modeling and prediction of ground motion parameters

In this study, the moment magnitude Mw is used to express the intensity at the source of anearthquake. Magnitudes are sampled independently of the time of occurrence of each earthquake

Copyright q 2008 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2009; 38:887–905DOI: 10.1002/eqe

EFFECT OF CUMULATIVE SEISMIC DAMAGE 889

5 6 7 8 90

0.2

0.4

0.6

0.8

1

Moment magnitude (Mw)F(

Mw

)

Figure 1. Cumulative density function for a=4.56 and b=0.91.

using a cumulative distribution function derived from the frequency–magnitude relationship givenby Gutenberg and Richter [12] as

Neq(Mw)=10a−bMw (1)

where Neq(Mw) is the cumulative annual rate of earthquakes having magnitudes greater than Mw,and a and b are dimensionless parameters that depend on the regional seismicity. The derivedsampling distribution is then

F(Mw)=1− 10a−bMw

10a−bMw,min(2)

where Mw,min is the smallest possible magnitude of earthquakes for the given region. As anillustration, Figure 1 shows the plot of F(Mw) for a=4.56 and b=0.91. These values are reportedby US Geological Survey [13] for the San Francisco Bay Area.

The occurrence of earthquakes is modeled as a Poisson’s process with a mean rate appropriatefor the region. The Poisson distribution is written as

f (x)= (�)x

x ! exp(−�) (3)

where x is the number of occurrences of an earthquake in the time window TH, which is thetime span over which LCC is computed, � is the mean number of earthquake occurrences in TH,and f (x) is the probability density function (PDF) of x . In Poisson’s process the time intervalsbetween two occurrences follow an exponential distribution. Therefore, the time of occurrences ofthe (M+1)th earthquake is simulated as follows:

tM+1= tM +�t, M=1,2,3, . . . (4)

where tM is the time of occurrence of the M th earthquake and �t is the time interval between twoearthquakes simulated using the following PDF:

f (�t)=(

�

TH

)e−(��t/TH) (5)



The peak ground acceleration AH and peak ground velocity VH at the bridge site are computedusing the ground motion attenuation relationships given by Campbell [14]. The attenuation rela-tionship for AH is written as

ln(AH) = −3.152+0.904Mw−1.328 ln√R2SEIS+[0.149 exp(0.647)]2

+[1.125−0.112 ln(RSEIS)−0.957Mw]F+[0.440−0.171 ln(RSEIS)]SSR+[0.405−0.222 ln(RSEIS)]SHR+�A (6)

where RSEIS is the distance of the source from the site of the bridge, F is an index variableused to indicate the style of faulting, SSR and SHR are index variables that represent the local siteconditions. The term �A is the model error that is modeled as a random variable having normaldistribution with mean of zero and standard deviation given by Equation (7)

�A={0.889−0.0691Mw, Mw<7.4

0.38, Mw�7.4(7)

The attenuation relationship for VH is given by

ln(VH) = ln(AH)+0.26+0.29Mw−1.44 ln[RSEIS+0.0203 exp(0.958Mw)]+1.89 ln[RSEIS+0.361 exp(0.576Mw)]+(0.0001−0.000565Mw)RSEIS−0.12F−0.15SSR−0.30SSR

+0.75 tanh(0.51D)(1−SHR)+ fv(D)+�V (8)

where �V is again the model error modeled as a normal random variable with zero mean andstandard deviation given by Equation (9)

�V=√

�2A+0.062 (9)

The function fv(D) in Equation (8) is given by Equation (10)

fv(D)={0, D�1km

−0.30(1−SHR)(1−D)−0.15(1−D)SSR, D<1km(10)

where D is depth to the base rock from ground surface at the bridge site. Figures 2 and 3 showone possible realization of the ground motion parameters AH and VH for TH=75 years and therandom variables shown in Table I.

2.2. Spectral acceleration and seismic energy demand

In the proposed formulation, the spectral acceleration, Sa, and the seismic energy demand, EI,are needed for each simulated earthquake. The spectral acceleration Sa is computed by scalingAH following Kunnath and Chai [15]. This formulation to compute elastic response spectrum,originally proposed by Vidic et al. [16] and later modified by Chai et al. [17], can be written as

Sa=�aAH (11)



Figure 2. A realization of AH values for TH=75 years.

Figure 3. A realization of VH values for TH=75 years.

Table I. Statistical parameters for seismicity modeling.

Variable Distribution Parameters Range

Mw See Equation (2) a=4.56Mw�5.5

b=0.91RSEIS Uniform 75km�RSEIS�100kmD Uniform 3km�D�6kmF Bernoulli p=0.5 F=0,1

x Poisson v=150x�0

TH=75years

where �a is a scaling factor defined as

�a=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

1+2.5(Ca−1)T

Tc, 0<T�0.4Tc

Ca, 0.4Tc<T�Tc

2�CvVHAHT

, T<0.4Tc

(12)



where T is the fundamental period of the bridge (or the equivalent SDOF system), Tc is thecharacteristic period of ground motion, Ca is the ratio of elastic spectral acceleration to peakground acceleration in the short period range, and Cv is the ratio of spectral velocity to peak groundvelocity in the velocity-controlled region of the response spectrum. The values suggested by Chaiet al. [17] for Ca and Cv are 2.5 and 2.0, respectively. The value of Tc is given by Equation (13)

Tc=2�Cv

Ca

VHAH

(13)

The seismic energy demand EI is defined by Kunnath and Chai [15] asEI= 1

2mv2e (14)

where ve is the equivalent input energy velocity given by Equation (15)

ve=�vVH (15)

where �v is a velocity amplification factor defined as

�v=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

�∗v

[2T

Tc−

(T

Tc

)2]

, T�Tc

�∗v

[T

Tc

]−�

, T>Tc

(16)

where �∗v is an energy amplification factor given as

�∗v= 0.25AH

VH

√tdTc

√�+0.5

2�+2(17)

where � is an input energy spectrum parameter equal to 0.5 as suggested by Kunnath and Chai [15],and td is the duration of strong ground motion. The value of td is given by Trifunac and Brady[18] as follows:

td=−4.88s+2.33Mw+0.149RSEIS (18)

where s is a geologic site parameter and is equal to 0.0, 1.0, and 2.0 for alluvium, intermediate,and rock, respectively. Note that the type of soil used for the numerical example shown later inthis paper is alluvium.

2.3. Peak displacement demand

The quantities Sa and T are used to compute the peak displacement demand Umax based on theprobabilistic demand model developed by Gardoni et al. [19] as

ln

(Umax

H

)=0.61+3.90��2+(1+��2)d̂�+�� (19)

where H is the clear height of the column, ��2 is a model parameter equal to −0.153 and�� =0.216. The variable d̂� is the natural logarithm of the deterministic drift demand computed



using a deterministic procedure originally proposed by Chopra and Goel [20] for the case ofbuildings and later modified by Gardoni et al. [19] for the case of bridges, and �� is a randomvariable that has standard normal distribution.

2.4. Number of inelastic cycles

For a given earthquake response, the equivalent number of constant amplitude inelastic cycles Ncorresponding to a certain amplitude is needed to compute the seismic damage. The value of Ncorresponding to Umax is obtained from cyclic demand spectrum [15] as

N = EI

4hVyUmax(20)

where Vy is the lateral force at yield and the parameter is the ratio between the hysteretic energyand the seismic energy demand, EI, that can be written as

=1.13(−1)0.82

(21)

where is the ductility demand given by

= Umax

Uy(22)

whereUy is the displacement at yield of the column top. The parameter h is a coefficient suggestedby Kunnath and Chai [15] to account for the deterioration of stiffness due to cyclic loading and isequal to 0.5.

3. CUMULATIVE SEISMIC DAMAGE

Under earthquake loading, bridge columns undergo several cycles of inelastic deflections. There-fore, low-cycle fatigue analysis is used in this paper to evaluate the seismic damage. In addition,an approximate strength degradation equation suggested by Das et al. [21] is used to computethe structural properties of the damaged structure. This section first presents the background andthe method adopted to model the low-cycle fatigue. Then, the computation of damage index (DI)is discussed. Lastly, the methodology to compute structural properties of a damaged structure ispresented.

3.1. Low-cycle fatigue

Based on Coffin [22] and Manson [23], the Coffin–Manson theory of fatigue formulates thebehavior of longitudinal bars under reversed cyclic loading as

�p=�′f(2Nf)c (23)

where �p is the plastic strain amplitude, �′f and c are material constants determined experimentally,2Nf is the number of half cycles for the first fatigue crack on the longitudinal reinforcement bar.



Mander et al. [24] obtained the following expression for �p based on experiments on reinforcementbars:

�p=0.08(2Nf)−0.5 (24)

Similarly, Kunnath et al. [10] obtained the following expression:

�p=0.065(2Nf)−0.436 (25)

Tsuno and Park [25] carried out an experimental work and reviewed the damage models developedby Mander et al. [24] and Kunnath et al. [10]. Tsuno and Park [25] tested five RC columnswith different loading patterns and compared the observed damage with the predicted damage.It was observed that Kunnath’s model predicts failure well for RC columns that are seismicallydesigned according to Caltrans or AASHTO to have a dominant flexural failure mode. Kunnath’smodel was based on experiments on RC columns and thus accounts for damage in columns as acomposite of steel and concrete. Mander’s model was based on experiments on steel reinforcementand accounted for fatigue in the steel only. However, one disadvantage of Kunnath’s model is thatit underestimates the damage in extreme loading cases that have large displacements in the firstcycle. Mander’s model was found to be more accurate in such extreme loadings than Kunnath’smodel. In this study, instead of the relation between �p and Nf (i.e. Equations (23)–(25)) we use amodification of the relation suggested by Kunnath and Chai [15] to model the low-cycle fatiguebehavior for circular ductile RC columns as follows:

Nf=(c1

)c2(26)

where Nf is number of cycles to failure corresponding to the ductility demand , c1 is a constantequal to 8.25 and c2 is a constant equal to 4.0. The above expression can be used only for thefirst earthquake and has to be modified for the future earthquakes because the column deteriorateswith every experienced earthquake. We propose the following variation of Equation (26) to makeit applicable to damaged columns:

NfM =(8.25

)4

−M−1∑i=1

Ni , M=2,3,4, . . . (27)

where NfM is the number of cycles to failure for the M th earthquake and Ni is the number ofcycles in the i th earthquake that preceded. The expression in Equation (27) can be explained usingFigure 4. If Ni is the number of cycles used up in the i th event, then Ni has to be subtracted fromthe column capacity for the i th event to obtain the deteriorated capacity for the (i+1)th event.Thus, the fatigue curve after the i th event is translated by the amount Ni toward the left.

3.2. Cumulative DI

Using the well-known Miner’s rule [26], the cumulative DI can be written as follows:

DI=m∑j=1

1

2Nf j(28)



Figure 4. Updating fatigue curve.

where DI is the cumulative DI after m half-cycles, Nf j is the number of cycles to failure corre-sponding to the displacement in the j th half-cycle of loading. Equation (28) can be modified tocompute the cumulative seismic DI after the M th earthquake as follows:

DIM = NM

NfM+DIM−1, M=2,3,4, . . . (29)

where NM is the equivalent number of constant amplitude inelastic cycles in the M th earthquakecomputed using Equation (20) and NfM is the number of cycles to failure for the peak displacementof M th earthquake computed using Equation (26). Based on the Miner’s rule, failure is assumedto occur when DI�1.0.

3.3. Structural properties of damaged structure

The structural properties xp of a pristine bridge are defined as follows:

xp=(k,T,Uy,Vy) (30)

where k is the lateral column stiffness. The vector xp in Equation (30) is represented by x−pM right

before the M th earthquake and by x+pM right after the M th earthquake.

x−pM = (k−

M ,T−M ,U−

yM ,V−yM ), M=1,2,3, . . . (31)

x+pM = (k+

M ,T+M ,U+

yM ,V+yM ), M=1,2,3, . . . (32)

Owing to structural damage, it is expected that the column stiffness k decreases due to eachexperienced earthquake. The cyclic loading during each earthquake also results in the failure of thebond between steel and concrete, which results in a larger displacement at yield Uy. Das et al. [21]suggested that Equations (33) and (34) account for any change in the stiffness and displacementat yield due to an earthquake

k+M =k−

M

[1−Umax M −U−

yM

Uu−U−yM

]0.1

, M=1,2,3, . . . (33)



where k−M and k+

M are column stiffness right before and after the M th earthquake. The quantityUu is the maximum displacement under monotonic loading of the pristine column. The yielddisplacement U+

yM after the M th earthquake is given by the following equation:

U+yM =

[k+k−

M

k+k+M

]U−yM , M=1,2,3, . . . (34)

where k is the pristine column stiffness. As expected, these equations show that the earthquakeloading decreases the column stiffness and increases the displacement at yield. The values of T+

Mand V+

yM can be found from following equations:

T+M = 2�

√m

k+M

, M=1,2,3, . . . (35)

V+yM = k+

MU+yM , M=1,2,3, . . . (36)

As an illustration, a basic Monte-Carlo simulation is performed using the random variables listedin Table I and the structural parameters listed in Table II to compute the failure probabilities atvarious time instances during the service life of an example bridge, typical of current California’spractice. As expected it is found that the failure probability increases with the age of the bridgedue to the damage accumulated during past earthquakes (Figure 5).

4. EFFECT OF CORROSION

The paper uses a corrosion equation based on Fick’s law of chloride ion diffusion that was originallydeveloped for uncracked media. While in the literature models for chloride ingress into cracks areavailable [6, 27], they rely on information regarding crack width and density that is not available incase of cumulative seismic damage. Furthermore, the models that account for cracking have beendeveloped for beams and are not applicable to columns since they do not account for the crackclosure due to the axial compression and can overestimate the chloride ingress. To avoid addinguncertainties in the modeling of the crack propagation, the formulation presented in this paperuses the corrosion equation based on Fick’s law. This section presents the methodology to predict

Table II. Structural properties of the example bridge.

Quantity Symbol Value

Axial load on column P 0.1Ag f ′c

Height of column H 4 000mmDiameter of column D 1 500mmDiameter of longitudinal bars db0 25mmArea of longitudinal bars �l 2.0%Diameter of transverse reinforcement ds 9.5mmSpacing of transverse reinforcement sv 60mmNominal strength of concrete f ′

c 36MPaNominal yield strength of steel fy 475MPa



10 20 30 40 50 60 700

0.5

1

1.5x 10-3

Time (years)

Fail

ure

Pro

babi

lity

Figure 5. Effect of cumulative seismic damage on annual failure rate. Numerical error of simulation=2%.

the corrosion initiation following Choe et al. [11] and the computation of structural properties ofthe corroded structure.

4.1. Corrosion initiation and structural properties of corroded structure

Corrosion is initiated in the steel reinforcement when the chloride concentration exceeds a criticalvalue Ccr. The corrosion initiation time Tcorr is given by Dura-Crete [28]. The reduction in steelreinforcement area due to corrosion is given by Choe et al. [11] based on the work of Vu andStewart [29] as follows:

db(t |Tcorr) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

db0, t�Tcorr

db0− 1.0508(1−w/c)−1.64

dc(t−Tcorr)

0.71, Tcorr<t�Tf

0, t>Tf

(37)

Tf = Tcorr+db0

{dc

1.0508(1−w/c)−1.64

}1/0.71

(38)

Here the reinforcement bar diameter db is expressed as a function of time t , the corrosion initiationtime Tcorr, the initial diameter db0, the cover depth dc, the water to binder ratio w/c, and the timeTf when, in theory, db reaches zero. The value of db at time t is computed using the value of Tcorr,which in turn is simulated using Equation (37). A sensitivity analysis with respect to the modelparameters can be found in Choe et al. [11].

For each realization of the life span in the Monte-Carlo simulation, one value of Tcorr issimulated along with a set of values of the ground motion parameters and the time of occurrencestM to completely represent a possible scenario of the seismic and corrosive environment. Whilethe ground motion parameters are used to compute seismic damage, Tcorr is used to computedb(tM |Tcorr) and then to compute the structural properties of the corroded structure. The vectors



x−pM and x+

pM−1 in Equations (31) and (32) are equal when corrosion is not initiated (i.e. t<Tcorr).

After the initiation of corrosion, x−pM is computed as follows:

T−M = T ′

M +M−1∑i=1

�Ti , M=2,3,4, . . . (39)

�Ti = T+i −T−

i (40)

where T ′M is the fundamental period at time tM (the time of occurrence of the M th earthquake)

of the structure deteriorated only due to the corrosion. As illustrated in Figure 6 above, T ′M does

not include the effect of the seismic damage. The value of T ′M can be found by computing the

column stiffness from a moment curvature analysis using the reduced reinforcement area at timetM obtained from Equation (37). In Figure 6, the curve A′B′ represents the deterioration in thefundamental period due to corrosion only. The curve AB takes into account the deterioration dueto both the seismic damage and the corrosion. It is assumed that AB is parallel to A′B′ at adistance equal to the sum of the �Ti due to each past earthquake. This assumption implies thatthe corrosion rate is independent of the seismic damage, which is reasonable because thoughlocalized corrosion might be accelerated near the cracks caused by earthquakes, the rate of theuniform corrosion in the column is unaffected. Similarly, the displacement at yield U−

yM can bewritten as:

U+yM =U ′

yM +M−1∑N=1

�UyN , M=2,3,4, . . . (41)

�UyN =U+yN −U−

yN (42)

where U ′yM is the yield displacement at time tM due to the corrosion only. Figure 7 compares the

contributions of the cumulative seismic damage and corrosion in the deterioration of the examplebridge.

5. LIFE-CYCLE COST ANALYSIS

The LCC of a structure is defined as the total cost that is incurred, or may be incurred, throughout itslife-cycle. LCC analysis provides a framework to support decisions about resource allocation relatedto the design, construction, and operation (inspection and maintenance). Different formulationsfor the LCC can be used, for instance, by including multiple limit states or considering severalhazards [5, 30–32]. In this paper, following, Wen and Kang [30] and Kong and Frangopol [9], theexpected LCC of a structure is expressed mathematically as follows:

E[LCC(p)]=E[CC(p)+NPV(CM(p), t)+NPV(CIN(p), t)+NPV(CF(p), t)] (43)

where CC is the initial construction cost, CM the maintenance costs, CIN the cost of inspections,CF the failure costs, E[·] indicates the expectation operator, and p=a vector of parameters thatdescribes the decision variables, which include aspects such as design criteria (e.g. structuraldimensions, peak ground acceleration) and inspection and maintenance schedules. The net present



Figure 6. Deterioration in fundamental period due to earthquakes and corrosion.

10 20 30 40 50 60 700

0.5

1

1.5x 10-3

Time (years)

Failu

re p

roba

bilit

y

with corrosion

without corrosion

Figure 7. Effect of cumulative seismic damage and corrosion on annual failure probability.Numerical error of simulation=2%.

value (NPV) is used to discount the cost to make decisions at time t=0. The NPV of a cost Cincurred at time t is computed as

NPV(C, t)= C

(1+r)t(44)

where r is the discount rate. The selection of appropriate discount rates for infrastructure systemsis a matter of ongoing debate and is beyond the scope of this paper, therefore, standard discountrates will be used (i.e. r<0.05).

5.1. Cost of construction CC

The initial construction CC can be partitioned into the construction costs of the piers Cpier, thedeck slab Cdeck, and the piles Cpile. Therefore, CC can be written as

CC=Cpier+Cdeck+Cpile (45)



The value of Cpier is computed as

Cpier= �d2HCuc

4+ �[�ld2+�s(D−2dc)2]H�sCus

4(46)

where Cuc is the cost per unit volume of the concrete work in columns and Cus is the cost per unitweight of steel reinforcement work. The unit costs Cuc and Cus include the cost of material, labor,and equipment used in the construction. The parameter �l is the area ratio of the longitudinal steelto the gross column area, �s is the volumetric ratio of the shear reinforcement to the column core,and �s is the weight density of the steel. The value of Cdeck is computed as the total area of thedeck multiplied by a cost per unit area of the deck. Finally, the value of Cpile is computed as thetotal pile length multiplied by a cost per unit pile length. In general, the quantities used to defineCC can be random. Therefore, the expected value of CC, E[CC], should be used in Equation (43)to compute E[LCC].

5.2. Costs of maintenance CM and inspection CIN

In the literature different cost functions have been proposed to model the maintenance cost (e.g.Kong and Frangopol [9], Mori and Ellingwood [8], and Ellis et al. [33]). In this paper, the followinglinear cost function is used:

CM=CC�Pf(t) (47)

where �Pf(t)= Pf(t−)−Pf(t+) is the reduction in the annual probability of failure at time t dueto the maintenance intervention. The cost of inspection CIN is assumed to be constant [8]. Theexpected value of the NPV of the total maintenance and inspection costs, E[NPV(CM+CIN, t)],which represents the investment of the owner in the bridge’s maintenance and inspection, is given by

E[NPV(CM, t)]+E[NPV(CIN, t)]=E

[m(TH)∑i=1

(CC�Pf(ti )+CIN)

(1+r)ti

](48)

where m(TH) is the number of maintenances within TH, and ti is the time at i th maintenance.

5.3. Cost of failure CF

The cost of failure CF is the sum of the cost to the owner of the structure and the user costs. Thecost to the owner is the direct cost of rebuilding the structure, which includes dismantling the oldstructure and building the new one. The user costs are associated with the indirect losses due tothe disruption of the activities that rely on the use of the bridge. Stewart and Val [5] assumed CFincluding all direct and indirect costs due to collapse of the structure to be 10 times the constructioncost (i.e. CF=10CC). In this paper, it is assumed that the bridge is not critical for the transportationnetwork and alternative routes are available for the users. Thus, the user costs are not included inthe LCC analysis and CF is assumed to be equal to 2CC. Therefore, the expected NPV of CF canbe computed as

E[NPV(CF, t)]=∫ TH

0

CF

(1+r)tPf(t)dt=

∫ TH

0

2CC

(1+r)tPf(t)dt (49)



5.4. Expected LCC

Using Equations (43)–(49), E[LCC] can be written as

E[LCC(p)]=CC(p)+E

[m(TH)∑i=1

(CC(p)�Pf(p, t)+CIN(p))

(1+r)ti

]+

∫ TH

0

2CC(p)

(1+r)tPf(p, t)dt (50)

In this paper, the following assumptions are made to compute E[LCC]: (1) a maintenance actionbrings the structure back to its original condition; (2) maintenance is performed at every inspectiontime; and (3) in the considered time horizon TH the probability of more than one failure is negligible.This assumption is made because it is more likely that the first failure occurs near the end of theTH and assuming that the structure is reconstructed after failure, the probability of subsequentfailures within TH is negligible.

5.5. Numerical examples

To illustrate the relation between LCC and the design parameters, p, the LCC analysis is first carriedout ignoring CM and CIN in Equation (50) and varying the column diameter D. Table III providesthe values of all the parameters used in the computation. All the construction costs are computeddeterministically using the unit construction costs provided by Caltrans contract cost data [34].Given that a closed-form solution of Equation (50) cannot be obtained, Monte-Carlo simulationsare used to compute the statistics of LCC. In the simulation, random sequences of earthquakes arerandomly generated and used to compute the costs incurred throughout the structure’s life-cycle.Figure 8 shows that as D increases, CC increases and E[NPV(CF, t)] decreases. It is also shownthat as the column diameter increases, the probability of failure reduces and the variation aroundthe mean LCC becomes smaller converging to the initial construction cost. The minimum valuefor E[LCC] is reached at D=1000mm. All the costs in Figure 8 are normalized to correspond toone million dollars of CC for D=1500mm.

The second numerical example explores the effects of CM and CIN on LCC of the bridgeshown in Table II. Three maintenance policies are investigated: (a) regular maintenance intervals,(b) random maintenance intervals with constant rate of occurrence, and (c) random maintenanceintervals with increasing rate of occurrence proportional to the annual failure probability. Monte-Carlo simulations are used to compute the statistics of LCC. Figure 9 shows the effect of thethree maintenance policies on the costs incurred in operating a deteriorating bridge through itsservice life. All the components of LCC normalized to one million dollars of CC are plottedagainst E[NPV(CM+CIN, t)]. Although CF in case when failure occurs is larger than CM and

Table III. Parameters used in computing life-cycle cost.

Item Cost

Steel work $2.25/kgConcrete work $800/m3

Cdeck per unit area $350/m2

Cpile per unit length $250/mLength of bridge 2 spans of 20m eachWidth of bridge 10mPile depth 15mr 2%



800 1000 1200 1400 1600 18000

2

4

6

8

10

12

14x 105

Column Diameter, D (mm)

Nor

mal

ized

Cos

t($) E [LCC]

E[LCC ] - σ

E [LCC] + σ

CC

E[CF]

Figure 8. Variation of LCC with column diameter.

CIN, E[NPV(CF, t)] is found to be insignificant because Pf(t) is relatively small. It is found thatthere is no significant change in E[LCC] for different maintenance strategies. It is also seen thatin all the maintenance strategies the E[LCC] increased with increase in E[NPV(CM+CIN, t)].This implies that for the example bridge, which is not critical in the transportation network, andthe example environmental and seismic conditions, conducting maintenance and inspections is noteconomically justified.

6. CONCLUSIONS

A methodology is presented to include the effect of cumulative seismic damage in columns inthe LCC analysis of bridges. The methodology is extended to account also for the structuraldeterioration due to the corrosion of the steel reinforcement. The uncertainties in the groundmotion parameters, seismic demand on the bridge, and corrosion process are accounted for inthe methodology. The same methodology can also be used to account for the uncertainties in thedistance to the source and the style of faulting.

Using an example bridge typical of current California’s practice, it is shown that the failureprobabilities increase significantly over a bridge service-life. This is because of the damage accu-mulated during the repeated occurrence of small earthquakes that do not lead to a failure of thebridge. It is also found that, in seismic regions, the contribution of cumulative seismic damage ishigher than the one from corrosion. The developed methodology can be used in an LCC analysisto assess the optimal design parameters for a bridge. As a practical illustration, the proposedformulation is used to assess the LCC of the example bridge and find the optimal column diameter.The LCC analysis is also carried out to study the effect of different maintenance policies. It isfound that although the bridge is deteriorating, maintenance cost is higher than the expected failurecost because of the low probability of failure throughout its service life. For this reason, based on



E[LCC]

(a)

(c)

E[LCC]

(b)

104 105

105

102

103

104

105

106

107

Nor

mal

ized

Cos

t ($)

102

103

104

105

106

107

Nor

mal

ized

Cos

t ($)

102

103

104

105

106

107

Nor

mal

ized

Cos

t ($)

E [NPV(CM+CIN,t)] ($)

E [NPV(CM+CIN,t)] ($)105

E [NPV(CM+CIN,t)] ($)

E [NPV(CM+CIN,t)]

E [NPV(CM+CIN,t)][NPV(CM+CIN,t)]

E [NPV(CF,t)]

E [NPV(CF,t)]E [NPV(CF,t)]

E[LCC]

CC

CCCC

Figure 9. Effect of maintenance policy on bridge costs: (a) regular intervals; (b) random intervals withconstant rate and (c) random intervals, rate proportional to failure probability.

the LCC analysis, conducting maintenance and inspections might not be economically justified forbridges that are not critical in the transportation network.

The results for the numerical example are specific to the example bridge for the chosenseismic scenario and environmental conditions. However, the observations are likely to be validalso for other similar bridges, scenarios, and conditions. Furthermore, the corrosion equationused for the analysis is based on Fick’s law of chloride ion diffusion that was originally devel-oped for un-cracked media. Therefore, the potential acceleration of the corrosion process dueto the presence of cracks associated with the cumulative seismic damage is not accounted for.



Nevertheless, the results are helpful to better estimate the reliability and assess the LCC of RCbridges.

ACKNOWLEDGEMENTS

The authors wish to thank Professors Joseph M. Bracci and John Mander of the Zachry Departmentof Civil Engineering and Professor Colleen Murphy of the Department of Philosophy at Texas A&MUniversity for helpful discussions and suggestions.

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Effect of cumulative seismic damage and corrosion on the life-cycle cost of reinforced concrete bridges