Structural Safety 31 (2009) 483–489

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Structural Safety

journal homepage: www.elsevier .com/locate /s t rusafe

Lifetime reliability-based optimization of reinforced concrete cross-sectionsunder corrosion

Fabio Biondini a,*, Dan M. Frangopol b

a Department of Structural Engineering, Politecnico di Milano, P.za L. da Vinci, 32, 20133 Milan, Italyb Department of Civil and Environmental Engineering, Center for Advanced Technology for Large Structural Systems, Imbt Labs, Lehigh University, Bethelhem, PA 18015-4729, USA

a r t i c l e i n f o

Article history:Received 8 November 2008Received in revised form 4 March 2009Accepted 30 June 2009Available online 25 August 2009

Keywords:Concrete structuresDiffusionCorrosionLifetime performanceReliability-based designCellular automata

0167-4730/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.strusafe.2009.06.008

* Corresponding author. Tel.: +39 02 2399 4394; faE-mail addresses: biondini@stru.polimi.it (F. Bion

edu (D.M. Frangopol).

a b s t r a c t

This paper presents a lifetime reliability-based approach to the optimization of reinforced concrete (RC)cross-sections in an aggressive environment. The lifetime structural performance is evaluated by using ageneral methodology for time-variant analysis of RC structures subjected to diffusive attacks fromaggressive agents with corrosion of the reinforcement. The lifetime probabilistic optimization is formu-lated at the cross-sectional level and is aimed to minimize the material cost under a time-dependent con-straint on the structural reliability. The optimization problem is solved by combining a discrete gradient-based optimization method with a Monte Carlo simulation. The obtained results demonstrate that in alifetime-oriented design the amount and location of the steel reinforcement and the value of the concretecover play a crucial role for the optimal achievement of the desired lifetime structural performance.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction [2,4,6,7,10]. A life-cycle approach to structural optimization has

The design of concrete structures under chemical–physicaldamage phenomena is usually based on simple criteria associatedwith prescribed environmental conditions. Such criteria introducethreshold values for concrete cover, water-cement ratio, amountand type of cement, among others, to limit the effects of local dam-age due to carbonation of concrete and corrosion of reinforcement.However, the lifetime performance of concrete structures exposedto aggressive agents may significantly depend on the structuralgeometry and reinforcement layout. For this reason, a rational life-time approach to optimum structural design of durable concretestructures should lead to find design solutions able to comply withthe desired performance not only at the initial time of construc-tion, but also during the expected lifetime by taking into accountthe effects induced by unavoidable sources of damage underuncertainty.

Based on these premises, life-cycle approaches to structuraloptimization of deteriorating systems have been proposed to high-light the fundamental role played by the time-variant performancein the optimal maintenance planning and selection of the optimumstructural design [1–13]. Reliability-based life-cycle optimizationapproaches to deteriorating RC structures have been reported in

ll rights reserved.

x: +39 02 2399 4220.dini), dan.frangopol@lehigh.

been recently developed in a deterministic context for truss andframe structures composed by homogeneous members [9,13], aswell as for RC frames [11], by assuming material degradation lawsassociated with general deterioration processes. In this paper thedeterministic formulation is extended to a probabilistic contextto the minimum lifetime cost design of RC cross-sections subjectedto diffusive attacks from environmental aggressive agents, like sul-fate and chloride, which may cause a deterioration process withcorrosion of the steel reinforcement.

The evolution of the lifetime structural performance is evalu-ated by using a general procedure proposed in previous works[14–16]. The lifetime probabilistic optimization is formulated atthe cross-sectional level and is aimed to minimize the cost of thematerials, concrete and steel, under a time-dependent constrainton the structural reliability. The optimization problem is solvedby combining a discrete gradient-based optimization method witha Monte Carlo simulation. The role played by a lifetime approach tostructural optimization is shown by comparing the optimal solu-tions obtained with a classical time-invariant formulation, whichconsiders the initial undamaged state only, and the proposed life-time formulation, where the time evolution of the structural per-formance is taken into account. The obtained results show thatin a lifetime-oriented design the minimum feasible area of rein-forcement is not associated with the maximum depth of the steelbars over the concrete cross-section, as expected in a classicaltime-invariant approach, but the amount and location of the rein-forcement and the value of the concrete cover play a crucial role.

z2 (j =2)

(i+1)

(i)

(i−1)

1 (j =1)i) (i+1)i−1) (( z

Fig. 1. Pattern of cells involved in the evolutionary rule of a two-dimensionalcellular automaton.

484 F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489

2. Lifetime performance of RC structures

A lifetime performance analysis of RC structures in aggressiveenvironments should be able to account for both the diffusion pro-cess of the aggressive agents, like sulfate and chloride, and the cor-responding mechanical damage induced by diffusion, whichusually involves corrosion of the reinforcement [17–19].

2.1. Cellular automata simulation of the diffusion process

In the proposed approach, the diffusion process of aggressiveagents in concrete is described by the Fick’s laws of diffusion[20] and reproduced by using a special class of evolutionary algo-rithms called cellular automata [21]. In its basic form, a cellularautomaton consists of a regular uniform grid of cells with a dis-crete variable in each cell which can take on a finite number ofstates. During time, cellular automata evolve in discrete time stepsaccording to a set of local evolutionary rules. For the diffusionproblem in one-dimension, the discrete variable in the cell i refersto the concentration Ck

i = C(zi, tk) at point zi and time tk, and the fol-lowing evolutionary rule can be adopted [14]:

Ckþ1i ¼ /0Ck

i þ1� /0

2Ck

i�1 þ Ckiþ1

� �ð1Þ

where /0 2 [0;1] is a suitable evolutionary coefficient. To prove theequivalence between the evolutionary rule of the cellular automa-ton and the diffusion differential equation, the previous relationshipcan be rewritten in the following equivalent form [22]:

Cðz; t þ DtÞ ¼ /0Cðz; tÞ þ 1� /0

2½Cðz� Dz; tÞ þ Cðzþ Dz; tÞ� ð2Þ

which refers to a cellular automaton with grid dimension Dz andtime step Dt, and where zi = z, zi±1 = z ± Dz, tk = t, and tk+1 = t + Dt.By subtracting C(z, t) from both sides and dividing by Dt, we obtain:

Cðz; t þ DtÞ � Cðz; tÞDt

¼ 1� /0

21Dt½Cðz� Dz; tÞ � 2Cðz; tÞ þ Cðzþ Dz; tÞ� ð3Þ

or:

Cðz; t þ DtÞ � Cðz; tÞDt

¼ 1� /0

2Dz2

Dt½Cðzþ Dz; tÞ � Cðz; tÞ� � ½Cðz; tÞ � Cðz� Dz; tÞ�

Dz2 ð4Þ

where the right hand term has been multiplied and divided by Dz2.By denoting:

D ¼ 1� /0

2Dz2

Dtð5Þ

and taking the limit Dz ? 0 and Dt ? 0, the Fick’s diffusion equa-tion is finally obtained:

@Cðz; tÞ@t

¼ D@2Cðz; tÞ@z2 ð6Þ

Therefore, the diffusion process can be simulated according to aprescribed value of the diffusivity D by properly relating suchparameter to the grid dimension Dz and the time step Dt of theautomaton. This demonstration can be generalized to d-dimen-sions (d = 1, 2, 3), where the diffusion equation:

Dr2Cðz; tÞ ¼ @Cðz; tÞ@t

; r2 ¼Xd

j¼1

@2

@z2j

ð7Þ

can be reproduced by using the following evolutionary rule:

Ckþ1i ¼ /0Ck

i þ1� /0

2d

Xd

j¼1

Cki�1;j þ Ck

iþ1;j

� �ð8Þ

with:

D ¼ 1� /0

2dDz2

Dtð9Þ

Fig. 1 shows the pattern of cells involved in the evolutionaryrule for a two-dimensional cellular automaton (d = 2).

The value /0 = 1/2 is generally a proper choice to obtain a goodaccuracy of the automaton. Morever, a deterministic description ofthe local diffusion mechanism usually allows to evaluate the globaleffects of the diffusion process with adequate accuracy for designpurposes. However, it is worth noting that the stochastic effectsin the mass transfer associated to the local random variability ofmaterial diffusivity D can easily be taken into account by assuming/0 as random variable. In this way, by adopting different probabi-listic distributions for cracked and uncracked concrete, it is alsopossible to simulate local modifications in the rate of mass diffu-sion induced by cracking, which usually involve higher gradientof concentration and coupling effects between diffusion and dam-age [14].

2.2. Modeling of steel corrosion

Structural damage induced by diffusion may involve deteriora-tion of concrete and corrosion of reinforcement [18]. This studywill focus on the effects of corrosion on the structural performance.To this aim, a deterioration process with no damage of concreteand uniform corrosion is considered. As shown in [14], the percent-age loss of steel resistant area for a corroded reinforcement bar canbe effectively described by means of a dimensionless damage indexds which provide a direct measure of damage within the range[0;1]. The corrosion rate of steel depends on the concentration ofthe aggressive agent [19] and based on available data for sulfateand chloride attacks [23] a linear dependency can be approxi-mately assumed. The damage index dsm = dsm(zm, t) of a reinforce-ment bar m located at point zm = (z1m, z2m) over the concretecross-section is therefore correlated at each time instant t tothe diffusion process by assuming a linear relationship betweenthe rate of damage and the mass concentration C = C(zm, t) of theaggressive agent [14]:

@dsmðzm; tÞ@t

¼ Cðzm; tÞCsDts

ð10Þ

where Cs represents the value of constant concentration which leadto a complete damage (ds = 1) after the time period Dts. Since corro-sion does not occur until the accumulation of the aggressive agent

h

c

i=1

i=2

i=3

j=1 2 3 4

s s

s s s Grid of the Automaton

Aggressive Agent

C0

(b)(a)b

Fig. 2. Design model of the concrete cross-section. (a) Geometry of the cross-section and symmetric layout of the reinforcing bars (i, j), i = 1, . . . , 3, j = 1, . . . , 4.(b) Grid of the cellular automaton and location of the aggressive agent.

F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489 485

at the bar surface exceeds a critical threshold of concentra-tion Ccr [24], the initial condition dsm(zm, t0m) = 0 with t0m =max{t | C(zm, t) 6 Ccr} is assumed.

3. Lifetime approach to reliability-based structuraloptimization

3.1. Formulation of the optimization problem

The purpose of a lifetime design optimization process is to find avector of design variables x which optimizes the value of an objectivefunction f(x), according to both side constraints with bounds x� andxþ, and inequality time-variant behavioral constraints gðx; tÞ 6 0overaprescribedlifetimeT.Thedeterministic lifetimedesignoptimi-zation problem can be formulated as follows:

minx2D

f ðxÞ D ¼ fxjx� 6 x 6 xþ;gðx; tÞ 6 0; t 6 Tg ð11Þ

Reliability-based design is concerned with the evaluation of theprobability of failure:

PFðx; tÞ ¼ P½gðx; tÞ 6 0�; t 6 T ð12Þ

or the corresponding reliability index:

bðx; tÞ ¼ �U�1½PFðx; tÞ�; t 6 T ð13Þ

where U = U(�) is the standard normal cumulative probability func-tion. Therefore, the lifetime probabilistic design optimization prob-lem can be formulated as follows:

minx2D

f ðxÞ D ¼ fxjx� 6 x 6 xþ;bðx; tÞP �bðtÞ; t 6 Tg ð14Þ

where the lifetime target reliability �b ¼ �bðtÞ is in general time-var-iant since it reflects several factors which may change over time,including type and importance of the structure, possible failure con-sequences, warning of failure occurrence, and socio-economic crite-ria [25,26].

3.2. Lifetime optimization of RC cross-sections

Several quantities may be chosen as target requirements for theoptimal design of RC cross-sections. Since no maintenance inter-ventions are applied during the lifetime T, the objective functionadopted in this study is related to the cost of the materials, con-crete and steel:

f ðxÞ ¼ AcðxÞ þ vAsðxÞ ð15Þ

where AcðxÞ and AsðxÞ are the total area of concrete and steel,respectively, and v ¼ cs=cc is the ratio between the unit costs ofsteel, cs, and concrete, cc , respectively. Clearly, additional cost com-ponents, i.e. cost of formwork and so on, may be considered in thepresent formulation. However, it is worth noting that the selectedobjective function represents a consistent criterion to compare dif-ferent design solutions rather than the actual structural cost in astrict sense.

By denoting MR ¼ MRðx; tÞ the resistant bending moment of thecross-section, and MA ¼ MAðx; tÞ the acting bending moment, atime-variant behavioral design constraint is set at the ultimatelimit state in terms of safety factor H ¼ Hðx; tÞ as follows:

Hðx; tÞ ¼ MRðx; tÞMAðx; tÞ

P 1; t 6 T ð16Þ

If a lognormal distribution can be selected as appropriate modelfor MR and MA, with mean values lR and lA and coefficients of var-iation dR and dA, respectively, the probabilistic distribution of thesafety factor H is also lognormal with the following statisticalparameters [27]:

lH ¼lR

lAð1þ d2

AÞ dH ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

R þ d2A þ d2

Rd2A

qð17Þ

Based on this model, the time-variant reliability index can becomputed as follows:

bðx; tÞ ¼ kHðx; tÞfHðx; tÞ

ð18Þ

where kH and fH are, respectively, the mean and standard deviationof the normal random variable ln H:

kH ¼ ln lH �12

f2H f2

H ¼ lnð1þ d2HÞ ð19Þ

For RC cross-sections the vector of design variables x usually in-cludes continuous variables, related to the size and geometry of thecross-section, and integer variables, associated to the standardizedsteel bar diameters generally available for ordinary RC structures.In this study, the lifetime reliability-based optimization problemis solved by combining a generalized reduced gradient methodand a branch-and-bound method [28] with a Monte Carlo simula-tion [29] applied at each step of the solution process to evaluatethe time-variant statistical parameters lR and dR of the resistantbending moment MR over the lifetime T.

4. Application to a RC cross-section

The proposed formulation is applied in the following to the life-time reliability-based optimum design of both geometrical dimen-sions and reinforcement layout of a rectangular cross-sectionunder diffusive attack of aggressive agents.

4.1. Design model

The design model of the cross-section refers to the layout of thereinforcing steel bars shown in Fig. 2a. With reference to a widthb = 400 mm and a bar spacing s = 50 mm, and denoting h the heigthof the cross-section, h� = (h � c) the maximum bar depth, c theconcrete cover, and ;ij the diameter of the reinforcing bar (i,j), avector x ¼ ½xT

1 xT2 �

T collecting a set of six design variables, includ-ing two non negative continuous variables x1 ¼ ½ h� c �T and fournon negative discrete variables x2 ¼ ½ ;1 ;2 ;3 ;34 �T ; with ;ij ¼;i 8j 6 3, is assumed. The optimal values of the design variablesare searched considering the following side constraints:

h�min 6 h� 6 h�max ð20Þ

486 F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489

cmin 6 c 6 cmax ð21Þ

;i ¼ 0 8;i < ;min; i ¼ 1;2 ð22Þ

;i 6 ;max; i ¼ 1; . . . ;3 ð23Þ

;3 P ;min ð24Þ

;34 P ;min ð25Þ

where h�min ¼ 550 mm, h�max ¼ 750 mm, cmin ¼ 50 mm, cmax ¼100 mm, ;min ¼ 12 mm, and ;max ¼ 26 mm, with a discrete step sizeD; ¼ 2 mm between subsequent feasible diameters.

4.2. Diffusion process and corrosion damage

The aggressive agent is assumed to be located along the exter-nal boundary of the cross-section with prescribed concentrationC0, as shown in Fig. 2b. The diffusion process is studied by usingthe proposed cellular automata approach [14]. With reference toa diffusivity D = 9.5 � 10�12 m2/s, the cellular automaton is definedby a grid dimension Dz = 15.5 mm and a time step Dt = 0.1 years.To highlight the role of stochastic effects in the mass transfer,the results obtained by assuming the evolutionary coefficient /0

as deterministic or as random variable with symmetric triangulardistribution in the range [0;1], are compared. With reference to adesign solution with h* = 650 and c = 75 mm, the time evolutionof the diffusion process over a lifetime T = 50 years is describedby the maps of concentration C(z, t)/C0 shown in Fig. 3 for deter-

0.00 0.10 0.20 0.30 0.40 0.

5 years 15 years

(a)

5 years 15 years

(b)

Fig. 3. Maps of concentration C(z, t)/C0 of the aggressive agent after 5, 15, 25, and 5(b) Stochastic diffusion.

ministic mass transfer (Fig. 3a), and stochastic mass transfer(Fig. 3b).

The mechanical damage induced by diffusion is evaluated byassuming Ccr = 0, Cs = C0, and Dts = 50 years. This damage modelreproduces a deterioration process with severe corrosion of steel,as may occur for carbonated or heavily chloride-contaminated con-crete and high relative humidity, conditions under which the cor-rosion rate can reach values above 100 lm/year [19]. Fig. 4 showsthe time evolution of the damage indices of the steel bars associ-ated to the diffusion process shown in Fig. 3. It is worth noting thatfor the case investigated in this study both deterministic andstochastic mass diffusion lead to comparable values of the damageindices over the lifetime T, with a maximum difference Dds,max <0.03.

4.3. Time-variant reliability and its threshold

The time-variant reliability index b = b(t) is evaluated over alifetime T = 50 years by assuming the material strengths as randomvariables. The compression strength fc is modeled by a lognormaldistribution with mean value 35 MPa and standard deviation5 MPa. The steel strength fy is modeled by a lognormal distributionwith mean value 500 MPa and standard deviation 30 MPa. Thetime-variant resistant bending moment MR is computed by assum-ing for concrete in compression a stress block depth 0.8y over theneutral axis depth y, and an ultimate strain in compressionecu = 0.35%. A time-invariant acting bending moment MA with log-normal distribution defined by a mean value lA =300 kN m and acoefficient of variation dA = 0.15 is prescribed. Finally, by assumingthat the frequency of periodic inspections or monitoring activities

50 0.60 0.70 0.80 0.90 1.00

25 years 50 years

25 years 50 years

0 years from the initial time of diffusion penetration. (a) Deterministic diffusion.

0.00

0.20

0.40

0.60

0.80

1.00

0 10 20 30 40 50

Time, t [years]

Dam

age

Inde

x, δ

s

(1,1) (1,2) (1,3)

(2,1) (2,2) (2,3)

(3,1) (3,2) (3,3)

(3,4) Steel Bars (i,j)

Fig. 4. Time evolution of the damage indices ds of the steel bars. The symbol (i, j)refers to the steel bars of the design model shown in Fig. 2a.

F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489 487

will be progressively increased over the years to provide damagedetection and early warning of failure, the following bilinear func-tion �b ¼ �bðtÞ is adopted as time-variant reliability threshold:

=20 =50

(a)

(b)

χ χ

=20 =50χ χ

Fig. 5. Optimal design solutions (see also Table 1). (a) Time-invariant approach whichevolution of the structural performance is taken into account.

�bðtÞ ¼�b0 þ ð�b1 � �b0Þ t

t1; 0 6 t < t1

�b1 þ ð�bT � �b1Þ t�t1T�t1

; t1 6 t 6 T

(ð26Þ

with �b0 ¼ 4:0, �b1 ¼ 3:5, �bT ¼ 1:5, and t1 ¼ 30 years.

4.4. Discussion of the results

The minimum cost design of the RC cross-section depends onthe unit cost ratio v. The role of this key parameter is investigatedby solving the lifetime optimization problem for three differentcost scenarios with v = 20, v = 50, and v = 100. A Monte Carlo sim-ulation is performed on a sample of about 2000 time-variant cross-sectional analysis at each step of the solution process. The samplesize has been chosen so to achieve a stable estimation of the time-variant statistical parameters of the safety factor. To highlight therole played by a lifetime approach to structural optimization, Fig. 5and Table 1 make a comparison between the optimal solutions ob-tained respectively with a time-invariant formulation which con-siders the initial undamaged state only (Fig. 5a, Table 1a), andthe proposed lifetime formulation where the time evolution ofthe structural performance is taken into account (Fig. 5b, Table 1b).

With respect to the time-invariant formulation, the lifetime ap-proach leads to an increase of total cost by 16%, 14%, and 12%, forthe unit cost ratio v = 20, v = 50, and v = 100, respectively. Ingeneral, as expected, the optimal depth h� of the cross-section

=100

χ

=100χ

considers the initial undamaged state only. (b) Lifetime approach where the time

Table 1Optimal design solutions (see also Fig. 5). (a) Time-invariant approach which considers the initial undamaged state only. (b) Lifetime approach where the time evolution of thestructural performance is taken into account.

v h [mm] h* [mm] c [mm] ;1 [mm] ;2 [mm] ;3 [mm] ;34 [mm]

(a)20 600 550 50 � � 22 1650 610 560 50 � � 22 14100 665 615 50 � � 20 16

(b)20 600 550 50 16 20 12 1250 630 555 75 12 20 12 12100 735 635 100 12 16 12 12

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20 30 40 50

Rel

iabi

lity

Inde

x, β

Lifetime Design

Time-invariant Design Target Reliability

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20 30 40 50

Rel

iabi

lity

Inde

x, β

Lifetime Design

Time-invariant Design Target Reliability

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20 30 40 50Time, t [years]

Rel

iabi

lity

Inde

x, β

Lifetime Design

Time-invariant Design Target Reliability

χ =20

χ =50

χ =100

Fig. 6. Time evolution of the reliability index for the optimal design solutionsshown in Fig. 5.

488 F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489

increases with the increasing of the unit cost ratio v. However, itsvalue is not significantly affected by the type of time-invariant orlifetime formulation. The optimal values of the other design vari-ables are instead strongly related to the effects of damage. In fact,if damage is not considered, the minimum feasible steel area ofreinforcement is associated with the maximum depth of the steelbars (i.e. the layer i = 3), and the minimum concrete cover is se-lected since its value does not affect the resistant bending moment(Fig. 5a). On the contrary, if the lifetime structural performance isconsidered, the location of each layer of steel bars, as well as thevalue of the concrete cover, can play a crucial role in the definitionof the minimum feasible area of reinforcement. In fact, due to thedamage process induced by diffusion, a suitable balance betweenthe opposite trend in maximizing the bars depth and minimizingthe damage effects is required. The way to achieve such trade-offdepends on the cost scenario, since an increasing in the unit costratio v involves not only a reduction of the total area of steel rein-forcement, but also an improvement of the reinforcement protec-tion from the aggressive agent, that is obtained with higherdepth values for the steel bars with larger diameters and larger val-ues of the concrete cover (Fig. 5b). These results highlight that thereinforcement layout and the concrete cover may significantly af-fect the time evolution of structural performance, and they shouldbe considered as key factors for a lifetime optimum design.

Finally, Fig. 6 shows the time evolution of the reliability indexfor the optimal solutions shown in Fig. 5. It can be noted that a life-time optimization approach is required to satisfy the reliabilityconstraint bðtÞP �bðtÞ not only at the initial time, but over the pre-scribed lifetime T.

5. Conclusions

A lifetime approach to reliability-based optimization of RCcross-sections subjected to diffusive attacks from environmentalaggressive agents has been presented. The lifetime probabilisticoptimization has been formulated in a restrictive case in whichthe optimization consists of the minimization of the cost of atime-variant constraint on the structural reliability without con-sidering maintenance. The role played by a lifetime approach tostructural optimization has been shown by comparing the optimalsolutions obtained with a classical time-invariant formulation,which considers the initial undamaged state only, and the pro-posed lifetime formulation, where the time evolution of the struc-tural performance is taken into account. The obtained resultsshowed that in a lifetime-oriented design the minimum feasiblearea of reinforcement is not associated with the maximum depthof the steel bars over the concrete cross-section, as expected in aclassical time-invariant approach. In fact, the amount and locationof the steel reinforcement and the value of the concrete cover playa crucial role in the achievement of the desired lifetime optimalperformance.

F. Biondini, D.M. Frangopol / Structural Safety 31 (2009) 483–489 489

To emphasize the importance of a lifetime approach in struc-tural optimization, the present study focused on the minimum costdesign of RC cross-sections with respect to the ultimate limit stateunder the assumptions of steel corrosion and randomness of thematerial strengths only. Further studies are required to investigatethe effects of maintenance interventions on the optimum struc-tural design along the lines proposed in [3,7], as well as the roleof other objectives, such as lifetime cost, and design variables,including cross-sectional shape and bar spacing, design constraintsrelated to serviceability limit states, additional damage mecha-nisms, such as concrete deterioration, and other sources of uncer-tainty, including randomness of geometrical parameters, diffusivescenario, and damage rates. Also, research on determination ofoptimal target reliabilities for design and upgrading of concretestructures along the lines proposed in [30] is necessary. Finally, re-cent studies in the field of multi-objective optimization of deterio-rating structures are useful in supporting decisions when cost andperformance are in conflict [8,10,13].

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