Optimal Maintenance Plan for RC Members by Minimizing Life ...

Journal of Advanced Concrete Technology Vol. 6, No. 3, 469-480, October 2008 / Copyright © 2008 Japan Concrete Institute 469

Scientific paper

Optimal Maintenance Plan for RC Members by Minimizing Life-Cycle Cost Including Deterioration Risk Due to Carbonation Chien Kuo Chiu1, Takafumi Noguchi2 and Manabu Kanematsu3

Received 28 February 2008, accepted 15 August 2008

Abstract This paper presents various models of deterioration due to carbonation taking into consideration uncertainty factors to estimate the initiation and the rate of corrosion and to analyze the structural capacity and serviceability of RC members, i.e. shear capacity, bending strength and width of severe cracking or spalling of columns and beams with corroded rein-forcing bars, based on simple formulas formed through past experiments. Then, it goes on to propose a method for evaluating the failure and severe spalling/cracking probability during earthquakes and the deterioration risk of members in specified years from construction. Further, through application of an immune algorithm to the minimization of the life-cycle cost including deterioration risk, an optimal maintenance plan and semi-optimal solutions with a high diversity for reinforced concrete members can be found in a single analysis. Finally, a case study is conducted to establish the effectiveness of this system.

1. Introduction

Reinforced concrete (RC) structures are a very important asset in urban environments. In order to build a sustain-able urban environment and solve environmental prob-lems, it is necessary to develop maintenance technology to extend the life of deteriorating RC structures. RC structures deteriorate with time in a corrosive environ-ment. Concrete cracking, spalling of concrete covers, and corroded reinforcing steel are common signs of deterio-ration induced by carbonation or chloride attack that may affect both the safety and serviceability of RC structures. Generally, structural members, beams, or columns whose reinforcing steel components are corroded, are appro-priately repaired/retrofitted following investigation of the cause of corrosion. Moreover, owners of deteriorat-ing RC structures need to allocate significant and ever increasing financial resources for their repair, retrofit or replacement. Thus, a cost-effective solution can only be determined on the life-cycle cost (LCC) basis, i.e., by taking into account all costs associated with design, construction, maintenance, and possible damage modes of a structure.

Corrosion of reinforcing steel is the primary cause of deterioration of RC structures. A number of studies have considered the LCC of RC structures deteriorating due to corrosion. To determine optimal maintenance strategies,

Mori and Ellingwood (1994) and then Frangopol et al. (1997) minimized the expected LCC of RC structures while maintaining their probabilities of failure for ulti-mate limit states below a certain specified value. Steward et al. (2003) and Val (2005) concentrated mainly on the effect of ultimate limit states and serviceability limit states associated with cracking and spalling of the con-crete cover. In the Val paper Val (2005), optimal times between repairs that minimize the total expected cost of failure have been calculated as a function of the ratio between the failure costs for ultimate and serviceability limit states. Nakamura et al. (2001) developed a bridge management system (J-BMS) that can be used to evalu-ate the serviceability of existing RC bridges and applied an immune algorithm (IA) and genetic algorithm (GA) to obtain optimal maintenance strategies by minimizing the cost of repair. However, few studies discuss how an optimal maintenance plan can be envisaged beforehand to minimize the LCC, including the deterioration risk induced by failure and severe spalling/cracking during earthquakes, of a RC building located in a corrosive environment with high hazard of earthquakes. Therefore, there is a need for a system that can be used to determine a maintenance plan to minimize the LCC in terms of structural capacity and serviceability in current society taking into consideration the high importance of per-formance design, life-cycle maintenance, and risk di-versification of RC structures (Kato et al. 2005). Based on the backgrounds given above, the purpose of this paper is to propose a system framework for determining an LCC minimizing maintenance plan of RC structures.

This paper describes a system that can estimate the LCC, including construction cost, repair cost, retrofitting cost, and the deterioration risk composed of the expected cost of failure and severe spalling/cracking during earthquakes, of RC structural members in a corrosive environment due to carbonation (Fig. 1). We also analyze

1Graduate Student, Dept. of Architecture, Graduate School of Engineering, The University of Tokyo, Japan.E-mail:[email protected] 2Associate Professor, Dept. of Architecture, Graduate School of Engineering, The University of Tokyo, Japan.3Assistant Professor, Dept. of Architecture, Graduate School of Engineering, Tokyo University of Science, Japan.

470 C. K. Chiu, T. Noguchi, and M. Kanematsu / Journal of Advanced Concrete Technology Vol. 6, No. 3, 469-480, 2008

the ability of an immune algorithm (IA) capable of searching for semi-optimal solutions with high diversity to find the optimal maintenance plan by minimizing the LCC. Moreover, in order to improve the computing ef-ficiency of the immune algorithm to find the optimal maintenance plan, a model of effectiveness of re-pair/retrofit strategies (three repair/retrofit strategies are proposed in this paper) is used to directly reflect the effects of maintenance strategies on failure probability and spalling probability. Finally, a case study is con-ducted to prove the effectiveness of this system.

2. Evaluation of deterioration risk and LCC

In this paper, the deterioration model of carbonation and the formulas for estimating the structural capacity of an RC member proposed in the design guidelines are used to evaluate the deterioration risk. Because these equations have been proved by many past experiments, the uncer-tainties are only focused on material deterioration and construction quality. 2.1 Deterioration model of carbonation When the pH of the pore water in the concrete sur-rounding the reinforcing steel decreases with the ingress of CO2, the protective oxide film on the steel is removed, opening the way for corrosion provided oxygen and moisture are present. In this paper, the process of corro-sion of reinforcing steel due to carbonation (Fig. 2) is modeled on the basis of previous research (Tottori et al. 2004). 2.1.1 Initiation stage This stage is defined as the time period required for cor-rosion to begin. Under steady hygrometric conditions, the depth of carbonation increases in proportion to the square root of time; therefore, the depth of carbonation, C, can be expressed in millimeters as follows:

tA=C (1)

where A = carbonation coefficient (mm/year0.5) and t = time of exposure (year).

Based on previous investigations, when the carbona-tion front reaches the surface of the steel, it can be as-sumed that corrosion has been initiated (AIJ 2004). This paper considers the uncertainties with regard to the car-bonation coefficient and concrete cover induced by en-vironmental or construction conditions to calculate the probability (Fig. 3) that the corrosion of the outermost reinforcing steel layer has been initiated by using Monte Carlo simulation. In the simulation, corrosion is consid-ered to have occurred when the probability of corrosion initiation exceeds 10%, with a level of confidence of 90% (failure probability of 10%) to be consistent with the prediction of corrosion initiation based on ASTM C876 1991. In the probabilistic analysis, the carbonation coefficient and concrete cover are treated as lognormal random variables with coefficients of variation of 45%

(Kawanishi et al. 2006) and 20% (Kato et al. 2005), respectively.

2.1.2 Propagation stage and former period of acceleration stage The propagation stage is defined as the period from cor-rosion initiation to cracking initiation of the concrete surface. The former period of the acceleration stage is defined as the period from crack initiation to spalling initiation of the concrete surface. Based on a developed previously model (Tottori et al. 2004), the rate of corro-

Condition Setting

Carbonation-InducedDeterioration

Structure Performance

Failure Probability

Deterioration Risk

Uncertainty:1. Concrete cover2. Rate of corrosion of steel3. Coefficient of carbonation

Truss-Arch theoryPlastic-Flexure theory

Earthquakes with a recurrence period of

500 years

Cost caused by failureCost caused by spalling

Serviceability Performance

Spalling Probability

Width of cracks exceeds 0.5 mm-1.0 mm

Spalling caused by cracks

Earthquakes with a reccurrence period of 100 years

Structure/member ConditionsEnvironmental Conditions

Fig. 1 Flowchart depicting how deterioration risk inducedby carbonation is evaluated.

Initiation Stage

Propagation Stage

Former period of Acceleration Stage

Latter period of Acceleration Stage

Service PeriodCorrosion Initiation

Spalling InitiationCracking Initiation

Fig. 2 Stages of deterioration caused by carbonation.

Fig. 3 Evaluation of the probability of corrosion initiation.

C. K. Chiu, T. Noguchi, and M. Kanematsu / Journal of Advanced Concrete Technology Vol. 6, No. 3, 469-480, 2008 471

sion in these two stages is assumed to be the same, and this rate is treated as a lognormal random variable with a mean equal to 0.06%/year (percentage weight loss) (Tottori et al. 2004) and a coefficient of variation of 50% (Takahashi et al. 2005). 2.1.3 Latter period of acceleration stage According to past research (Val et al. 2003), spalling is caused with the formation of a crack with a width in excess of 0.5–1.0 mm and the threshold of the latter period of acceleration stage is difficult to define. How-ever, it can be reasonably assumed that the concrete surface spalls severely so that the concrete cover is al-most unable to prevent the steel from corroding in the latter period of the acceleration stage. Therefore, in this paper, the latter period is characterized by severe spalling, which is regarded as an increase in the crack width in excess of 1.0 mm; the corrosion of steel as a percentage weight loss in a 1.0 mm crack defined as the threshold of the latter period of acceleration stage can be quantified by Eq. (2) and Eq. (3) proposed in past research (Ozaki et al. 2005).

x×)φ/d078.1exp(141.0=wc (2)

os

c

φρw400

=r (3)

where wc = corrosion of steel (mg/mm2), x = crack width (mm), d = concrete cover (mm), φ = diameter of steel (mm), φo = diameter of steel without corrosion (mm), ρs = density of steel (mg/mm3, 7.85), and r = corrosion of steel as percentage weight loss (%).

Based on previous research (Tottori et al. 2004) and (Takahashi et al. 2005), the rate of corrosion at this stage is equal to the rate of corrosion of steel without any concrete cover and is treated as a lognormal random variable with a mean equal to 0.14%/year (percentage weight loss) and coefficient of variation of 50%.

2.2 Carbonation suppressive effect of finishing Because finishing can prevent the concrete surface from coming into contact with CO2, it can also suppress car-bonation. Therefore, finishing could be a possible alter-native to provide higher building durability depending on what kind of materials are used in the process. In this paper, if the finishing material is composed of cement, the depth of carbonation can be estimated as shown in Eq. (4) based on the concept of an equivalent concrete cover (AIJ 2004).

eqM-tAC = (4)

where Meq = equivalent concrete cover in relation to the thickness of finishing (mm).

If the finishing material is made up of a polymer mix-ture, we must use Eq. (6)-(8), which include a material aging factor dependent on ultraviolet radiation, rain, and wind, to estimate the depth of carbonation based on the

general form (Eq. (5)) (AIJ 2004 and Kono et al. 2004, 2006).

)R-Rt(AC 2+= (5)

)(R-])R(t∆t[AC '1-i

2'1-i

'ii ++= (6)

2'1-i

2'1-i1-i

'1-i )R(-)RA/C(t += (7)

1-it090.0-r

'1-i e135.0TR ××= (8)

Conditions: 1t∆ = ， 0C0 = ， 0t0 =

where R = resistance of carbonation in relation to year and coating thickness (year0.5), Ci = depth of carbonation in step i (mm), ti-1

’ = equivalent time of exposure in step i-1 (year), Ri-1

’ = resistance of carbonation in step i-1 (year0.5), ti-1 = time of exposure of finishing (polymer mixture) in step i-1 (year), and Tr = depth of material of polymer mixture (µm). 2.3 Probability of failure and spalling 2.3.1 Capacity degradation of beam and column The reduction in the shear and flexure capacity of an RC beam or column due to corrosion is considered for safety performance under earthquake conditions.

In this paper, two major causes of the reduction in capacity due to corrosion are considered. The first cause is the degradation of the yield stress of the reinforcing steel. In the case of general corrosion, the yield stress of reinforcing steel, σy, can be estimated as (JCI 1998)

)100/r17.2-1(σσ yoy = (9)

where σyo = yield stress of reinforcing steel without cor-rosion (N/mm2) and r = corrosion of steel as percentage weight loss (%).

The second cause is the degradation of the ultimate bond stress of the reinforcing steel and concrete, τbu, and it can be estimated as (JCI 1998)

)r0607.0-exp(ττ bobu ×= (10)

Bbwbo σ)d/bp52.1(τ += (11)

where τbo = ultimate bond stress without corrosion (N/mm2), b = width of beam or column (cm), db = di-ameter of main bar (cm), pw = stirrup ratio and σB = compressive strength of concrete (N/mm2). (1) Shear capacity The shear capacity of an RC beam or column in the case of bonding failure or stirrup yielding induced by corro-sion can be estimated by the truss-arch theory associated with Eq. (9), Eq. (10) and Eq. (11), as follows.

The shear capacity in the case of bonding failure, Vbu, can be considered as two items (Eq. (12)) according to the truss-arch theory illustrated in Fig. 4 (AIJ 1998, 2004


and Chiu et al. 2007).

atbu VVV += (12)

The first item, Vt, is caused by the truss mechanism related with the stirrup and bonding between the concrete and main bar. If the bonding failure happens before stir-rup yielding, it can be estimated as

∑ L/)j-L(jφτV ttbut ×××= (13)

where L = clear span length of beam or column (cm), Σφ = sum of perimeter of main bars (cm) and jt = distance between compression and tension bars (cm).

The second item, Va, is caused by the arch mechanism related with concrete and can be estimated as

)2/D(bθtan)σ-σν(V ctBa ××= (14)

D/L-}1)D/L{(θtan 2 += (15)

where ν = effective coefficient of compressive strength of concrete, D = depth of beam or column (cm), σct = compressive stress of concrete due to truss mechanism (N/mm2) and θ = angle of arch mechanism.

The shear capacity in the case of stirrup yielding, Vu, can be estimated as

2/νσbD)β-1(θtanΦcotσpbjV Bwywtu += (16)

)νσ/(}σp)Φcot1{(β Bwyw2+= (17)

where Φ= angle of compressive strut (if assumed, 45o) and σwy = yield stress of stirrup (N/mm2). If corrosion is present, the yield stress can be estimated by Eq. (9). (2) Flexure capacity The flexure capacity can be estimated by the plastic flexure theory associated with the Euler-Bernoulli beam theory (Eq. (18) and Eq. (19)) and Eq. (9) (AIJ 1988, 2004 and Chiu et al. 2007).

dσa9.0M ytu = (Beam) (18)

)σbD

N-1(ND5.0Dσa8.0M

Bytu += (Column) (19)

where at = area of main bar (cm2), d = effective depth of beam or column (cm) and N = axial force of column (N). (3) Shear demand The shear demand of an earthquake with a recurrence period of 500 years is defined as the shear force caused by an earthquake and the vertical loading, QDS.

ELDS QQQ += (20)

where QL = shear force caused by vertical loading (N) and QE = shear force caused by an earthquake with a recurrence period of 500 years (N).

The hazard analysis or seismic design codes can be used to analyze the ground acceleration of a specified region under an earthquake with a recurrence period of 500 years and then QE can be estimated by the pushover method.

2.3.2 Structure performance and failure prob-ability Using the capacity and the shear demand of an RC beam or column obtained based on 2.3.1, the shear perform-ance index, DV , and the flexure performance index, DM, are defined as follows:

DS

ubuV Q

)V,Vmin(D = (21)

DS

uM Q

L/M2D = (22)

where L = clear span length of beam or column (cm). If DV ≤ 1.0, shear failure is implied. If DM ≤ 1.0, flexure failure is implied.

Clearly, DV and DM decrease and approach the value of the limit (equal to 1.0) with the passage of time, as il-lustrated in Fig. 5. In this paper, the rate of corrosion of reinforcing steel is treated as a random variable and Monte Carlo simulation is used to calculate the distribu-tions of DV and DM. Then, the area of the distribution below the limit value can be evaluated and defined in terms of the shear failure probability, PV, and flexure

Fig. 5 Definition of shear failure probability.

Vt

L-jt jt

Vt

τbu

τbu

s ct

main bars

main bars

Fig. 4 Truss mechanism of truss-arch theory.


failure probability, PM, respectively. In addition, the sum of both is defined as the failure probability of an RC beam or column.

2.3.3 Serviceability performance and spalling probability It is generally perceived that the serviceability per-formance of RC structures is affected when spalling is caused with the formation of a crack with a width in excess of 0.5–1.0 mm (Val et al. 2003). The limit quan-tity of corrosion resulting in spalling can be estimated by Eq. (2) and Eq. (3). Therefore, when the quantity of corrosion of the outermost reinforcing steel component in the RC beam or column exceeds the limit quantity, spalling can be assumed to occur on the surface of the concrete. Monte Carlo simulation is used to evaluate the spalling probability, PS, similar to how it was used in the case explained in 2.3.2. However, it is reasonable to expect that a higher spalling probability caused by car-bonation would lead to a higher severe spalling/cracking probability when an earthquake with a recurrence period of 100 years occurs. Therefore, the product of the spalling probability caused by carbonation and the an-nual probability of earthquake with recurrence period of 100 years is defined as the annual probability of ser-viceability failure resulting from severe spalling/ crack-ing in this paper. 2.4 Deterioration risk and life-cycle cost 2.4.1 Deterioration risk of service period In this paper, the deterioration risk of a specified service period, E, is estimated as

∫T

0sssfVMft dt)νPCν)PP(C(

)k1(1

E ××+×+××+

= (23)

where Cf = cost of failure, Cs = cost of severe spalling/cracking, vf = annual probability of earthquake with recurrence period of 500 years (if a Poisson distri-bution is assumed, 0.2%), vs = annual probability of earthquake with a recurrence period of 100 years (if a Poisson distribution assumed, 1%), T = specified service period (year) and k = discount rate.

Eq. (23) contains two important observations. First, the expected cost of failure for an ultimate limit state results from an earthquake with a recurrence period of 500 years. Second, the expected cost of serviceability failure results from severe spalling/cracking when an earthquake with a recurrence period of 100 years occurs. However, the cost of failure and the cost of severe spalling/cracking are composed of many kinds of loss, e.g. function loss, injuries and so on, and are not dis-cussed in this paper. According to previous research (Kanda et al. 1997; Stewart et al. 2003), they vary de-pending on the structure occupation type and can be normalized by the construction cost of the structure. In addition, according to past experiments for corro-sion-induced columns, when a crack with a width in

excess of 0.5-1.0 mm caused by corrosion occurs, the ductility ratio decreases to 1.1-1.5. In other words, duc-tility of a deteriorating column cannot be expected even if the flexure failure mode happens under an earthquake. In this paper, it is assumed that not only shear failure but also flexure failure will lead to a catastrophic cost, based on the concept of possible maximum loss assessment. Additionally, because the cost of a carbonated column caused by earthquakes is focused on in this paper, the cost caused by spalling resulting from corrosion expan-sion without any external loads is not included.

2.4.2 Life-cycle cost This paper uses the concept of the LCC to find the op-timal maintenance plan for an RC beam or column. Be-sides the deterioration risk stated in 2.4.1, the construc-tion cost, retrofitting cost, and repair cost must also be considered. Therefore, the LCC, CT, is defined as

E))k1(

1C()

)k1(

1C(CC

tRepRe ttRetpReIT ∑∑ ++

×++

×+= 　　 (24)

where CI = construction cost, CRep = repair cost, CRet = retrofitting cost, tRep = year of repair, tRet = year of retro-fitting. 3. Maintenance strategy

In order to find the optimal maintenance plan for an RC beam or column, it is necessary to define the re-pair/retrofit techniques that can be applied. The selection of repair/retrofit techniques significantly affects the life-time performance of an RC beam or column. In this paper, three repair/retrofit techniques are considered. Moreover, in order to improve the computing efficiency of the immune algorithm to find the optimal maintenance plan, the model illustrated in Fig. 6 is used to directly reflect the effect of the maintenance strategy on the failure probability and spalling probability instead of re-analysis of deterioration taking into consideration of update con-ditions due to the maintenance strategy by FDM (Finite Difference Method). TYPE-I: Finishing renewal. In this method, the finishing of an RC beam or column is renewed by using the original material. For this purpose, not only recoating the finishing material but also repair for spalling/cracking is carried out. This is done so that the carbonation sup-pressive effect can be restored to its initiation stage magnitude without aging. Because the carbonated area still exists following application of this strategy, the spalling probability can be set to zero and then the process can proceed along the original curve (no main-tenance) without an initiation stage. Due to the exclusion of steel supplements in this strategy, there is no any in-fluence on the failure probability. TYPE-II: Finishing renewal and patch repair. In this method, not only the carbonated area but also the fin-


ishing is replaced by using the original material. With this strategy, the spalling probability can be set to zero and then the process can proceed along the original curve (no maintenance). Because patch repair is also practiced, the carbonated area is removed and the degradation of the structure capacity can be halted for a particular time period, which is equal to the initiation stage. TYPE-III: Finishing renewal, patch repair, and steel supplementing. In addition to the operations performed in TYPE-II, the steel is supplemented in TYPE-III ac-cording to the quantity of corrosion of the outermost reinforcing steel component. With this strategy, the spalling probability and failure probability can be set to zero and then the process can proceed along the original curve (no maintenance). 4. Immune algorithm

The immune system (Luh et al. 2004) protects living bodies from the invasion of foreign substances such as viruses, bacteria, and other parasites (called antigens). The immune algorithm, IA, (Nakamura et al. 2001; Honma et al. 2005), as illustrated in Fig.7, which emu-lates the biological immune system, is mainly composed of the genetic algorithm (GA) used to reform solu-tions/antibodies of the next generation, memory cells used to conserve higher-affinity solutions/antibodies, and suppressor cells (T-cells) used to suppress the re-seeking of higher-affinity solutions conserved in memory cells and proposed to solve optimization problems. Therefore, analogous to the biological immune system, the immune algorithm is capable of seeking multiple feasible solu-tions while maintaining diversity by means of memory cells and suppressor cells. In addition, IA has higher computing efficiency than GA for multiple feasible so-lution seeking based on the past research.

For decision making in the maintenance plan of an RC

member or building, not only the optimal solution but also semi-optimal solutions are important. Consequently, IA is used to find multiple semi-optimal plans including the optimal one instead of GA in this paper.

4.1 Computing procedure In this section, we describe the algorithm illustrated by the flowchart shown in Fig. 8. 4.1.1 Recognition of antigen Antigens and antibodies correspond to objective func-tions and associated feasible solutions. In this algorithm, memory cells are used to conserve higher-affinity solu-tions/antibodies, which can be regarded as a set of fea-sible solutions. In addition, objective/antigen and solu-tions/antibodies are defined as the LCC (CT) and main-tenance plans of RC members in this research, respec-tively. 4.1.2 Randomly initialized antibody population Similar to the genetic algorithm developed by Kane-matsu (Kanematsu et al. 2001), the initial population of

Fig. 6 Setting of repair/retrofit.

Fig. 7 Concept of Immune algorithm.

Cells of generating

antibodies

remove/solveremove/solve

Antigen (objective function)Antibody(feasible solution)

Antibody

Memory cell

Suppressor cell

suppress

stimulate

BODY


antibodies utilizing a multi-variable binary-code string is generated randomly. In this paper, four bits are used to express one repair/retrofit work including the method (2 bits, 22 kinds) and time (2 bits, 22 kinds) with which these methods were planned, as illustrated in Fig. 9. Further, each maintenance plan has a maximum of ten re-pair/retrofit works; therefore, 40 bits are built in one antibody. 4.1.3 Calculating affinity of antibody-antigen or antibody-antibody In this paper, the inverse of the LCC is used to express the affinity of the antibody v and the antigen, as shown in Eq. (25). Clearly, the greater the affinity, the lower is the LCC caused by the antibody v and its higher grades. The affinity Ψvw between antibodies v and w is defined as the ratio of the similarity of binary code strings in them. Note that 0 ≤ Ψvw ≤ 1, and when Ψvw = 1, the antibodies v and w, are identical.

Tν C

1Φ = (25)

4.1.4 Calculating concentration of antibody The concentration Θv is used to express the similarity of the antibody v with other antibodies, as shown in Eq. (26).

∑N

1ωνων π

N1

Θ=

= (26)

where N = the number of antibodies. If Ψvw ≥ Tπ1, πvw = 1; otherwise, πvw = 0; Tπ1 = user-defined threshold value that illustrates the allowable difference between two antibodies.

In this algorithm, as the concentration increases, the higher probability is selected as the memory cell and suppressor cell.

4.1.5 Antibody differentiation into memory and suppressor cells If the concentration Θv of the antibody v exceeds the user-defined threshold value TΠ, the antibody v becomes a candidate memory cell. Then, candidate memory cells are chosen for differentiation into a pool of memory cells in the order of the affinity of antibody-antigen and into a pool of suppressor cells in the order of the concentration. 4.1.6 Stimulation or suppression of antibody promotion In this paper, we use the affinity of the similar antibodies. Therefore, if Ψsv exceeds the user-defined threshold value TS, the antibody will be eliminated. In addition, in order to add and maintain the diversity of the next gen-eration of antibodies, those antibodies that have a high expected value based on Eq. (27) are chosen to create the next generation by the GA method (including selection, crossover, and mutation).

∏=

=S

1svsνν )Ξ-1(ΦE (27)

where S = number of suppressor cells. If Ψvw ≥ Tπ2, Ξvw = Ψvw; otherwise, Ξvw = 0. 4.2 Diversity of solution/antibody (Honma et al. (2005)) In this paper, we use the concept of entropy to estimate the diversity of a solution/antibody. First, the diversity of the locus j of an antibody in the same generation Hj(N) can be expressed as shown in Eq. (28). Then, the average value of Hj(N) is estimated by Eq. (29) to represent the diversity of the generation. It could be said that the higher the H(N), the higher is the diversity.

∑L

1iijijj plogp-)N(H

== (28)

∑G

1jj )N(H

G1

)N(H=

= (29)

where N = number of antibodies in one generation, G = number of loci in one antibody, L = number of symbols in one locus, and pij = probability of the symbol i appearing in locus j.

Fig. 8 Flowchart showing computing procedure in IA.

Fig. 9 Code of an antibody.


5. Case study

The optimal maintenance plans defined as the cases of minimal LCC for the columns (outside column) coated with mortar (C1), those coated with resin (C2) and those without coating (NC) on the first, sixth and twelfth floors of a twelve-story RC building are analyzed by the method proposed in this research to prove its suitability.

The parameters required for the analysis of deteriora-tion are listed in Table 1 and Table 2 and those of the computing of the LCC and optimization are listed in

Table 3 (June et al. 1997; Mark et al. 2003) and Table 4 (Honma et al. 2005).

The LCC, failure probability, and spalling probability of the columns on the first and sixth floors are illustrated in Fig. 10 and Fig. 11 (NC_M, C1_M, C2_M represent maintenance cases). The optimal maintenance plans of columns without coating based on IA and GA are illus-trated in Table 5. According to the results, when the same finishing material is used, the spalling probability curve of the column without maintenance does not change with the floor considered, and these curves with maintenance on the first and sixth turn out to be different because of the difference between them in the optimal maintenance plan. However, because a higher floor has a lower shear demand, the failure probability of the col-umn without maintenance on the sixth floor is smaller than that on the first floor. After the optimal maintenance plan is practiced, the failure probability of the columns on the first and sixth floor can be decreased to below 10% and almost zero, respectively. In addition, in the case of the twelfth floor, the failure probability of the column is zero irrespective of maintenance.

The LCC, deterioration risk and rates of each cost in the LCC are illustrated in Fig. 12, Fig. 13 and Fig. 14. When compared with the LCC and deterioration risk without maintenance, the optimal maintenance plans

Table 1 Parameters for the estimation of deterioration of columns. Location of column First floor Sixth floor Twelfth floor

Depth of column (D, cm) 95 95 90 Width of column (d, cm) 95 95 90

Clear span length of the column (L, cm) 400 350 350 Thickness of concrete cover (x, cm) 4.00 4.00 4.00

Diameter of main bar (db, cm) 4.1 4.1 3.2 Yield stress of main bar (σyo, N/mm2) 400 400 400 Yield stress of stirrup (σwyo, N/mm2) 800 800 300

Stirrup ratio (pw) 0.003 0.003 0.003 Shear demand (QDS, kN) 2092 1001.4 196.8

Compression strength of concrete (kN/mm2) 25.0 Water-cement ratio (%) 60

Tensile reinforcement ratio (%) 2.5 Carbonation coefficient (mm/year0.5) 7.57

Table 2 Parameters of the materials for finishing. Thickness of mortar (mm) 20

Thickness of resin coating (mm) 0.10 Cost of mortar 0.05CI

Cost of resin coating 0.10CI Period of durability of resin coating (year) 10

Table 3 Parameters for the estimation of LCC.

Construction cost (CI) (without coating) 1.00 Cost of failure (Cf) (Jun et al. (1997)) 10.0 Cost of severe spalling/cracking (Cs)

(Mark et al. (2003)) 2.00

Finishing renewal 0.05 Finishing renewal and patch repair 0.15

Finishing renewal, patch repair, and steel supplement 0.30

Discount rate (k) 0.00

Table 4 Parameters for GA and IA. Number of antibodies 500 Number of generations 1000

Selection method Roulette and elite conservation

Crossover method Crossover of one point with probability of 75%

Probability of mutation 0.5% Number of memory cells (IA) 10

Tπ1 ,Tπ2 ,TS (IA) 0.80 (Honma et al. (2005))

TП (IA) 0.60 (Honma et al. (2005))

0 20 40 60 80 100Service Period (year)

0

20

40

60

80

100

Failu

re P

roba

bilit

y (%

)

NC(1F)C1(1F)C2(1F)NC_M(1F)C1_M(1F)C2_M(1F)

Fig. 10a Failure probability of columns on the first floor.


yield a definite and effective improvement, irrespective of which floor the column belongs to. Moreover, ac-cording to the results, it is said that the rate of deteriora-tion risk, if the optimal maintenance practiced, is de-creased to below 5% from almost 60% expect columns without coating and the rate of repair/retrofit cost is about 30% expect the column without coating on the first floor.

Furthermore, the diversity of the solution/antibody is used to prove the effectiveness of this system and is

shown in Fig. 15. In this figure, besides the diversity of the memory cells (if in GA, the best 10 are chosen) (SAH), the diversity of each generation is also expressed. With reference to the diversity of solution/antibody in one generation (SH), IA shows more intense variation than GA. Due to the high intensity variation of diversity, IA can suppress similar solutions and does not easily resort to global-solution searching. In other words, IA does not converge to one solution by the original ad-justment mechanism (suppressor cells). In addition, the


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Failu

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Failu

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)


Fig. 10b Failure probability of columns on the sixth and twelfth floors.


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Fig. 11 Spalling probability of columns on the first and sixth floors.

Table 5 Optimal maintenance plan (without finishing).

First floor (1F) CT Maintenance plan (Year-TYPE) GA (Optimal) 2.13 15-Ⅱ, 30-Ⅱ, 45-Ⅱ, 60-Ⅲ, 75-Ⅱ, 90-Ⅰ (minimal LCC) IA (Optimal) 2.13 15-Ⅱ, 30-Ⅱ, 45-Ⅱ, 60-Ⅲ, 75-Ⅱ, 90-Ⅰ (minimal LCC)

IA 2.14 15-Ⅱ, 30-Ⅱ, 45-Ⅱ, 60-Ⅰ, 65-Ⅲ, 80-Ⅰ, 85-Ⅱ IA 2.14 15-Ⅰ, 20-Ⅱ, 35-Ⅱ, 50-Ⅲ, 65-Ⅱ, 80-Ⅱ

Sixth floor (6F) CT Maintenance plan (Year-TYPE) GA (Optimal) 1.93 15-Ⅱ, 30-Ⅰ, 40-Ⅱ, 55-Ⅰ, 65-Ⅰ, 75-Ⅰ, 85-Ⅰ, 95-Ⅰ (minimal LCC) IA (Optimal) 1.93 15-Ⅱ, 30-Ⅰ, 40-Ⅱ, 55-Ⅰ, 65-Ⅰ, 75-Ⅰ, 85-Ⅰ, 95-Ⅰ (minimal LCC)

IA 1.93 15-Ⅱ, 30-Ⅱ, 45-Ⅰ, 55-Ⅱ, 70-Ⅰ, 80-Ⅰ, 90-Ⅰ IA 1.95 15-Ⅰ, 25-Ⅰ, 35-Ⅱ, 55-Ⅰ, 65-Ⅰ, 75-Ⅰ, 85-Ⅱ

Twelfth floor (12F) CT Maintenance plan (Year-TYPE) GA (Optimal) 1.60 20-Ⅰ, 30-Ⅰ, 45-Ⅰ, 55-Ⅰ, 65-Ⅰ, 75-Ⅰ, 85-Ⅰ (minimal LCC) IA (Optimal) 1.60 20-Ⅰ, 30-Ⅰ, 45-Ⅰ, 55-Ⅰ, 65-Ⅰ, 75-Ⅰ, 85-Ⅰ (minimal LCC)

IA 1.63 20-Ⅰ, 40-Ⅰ, 50-Ⅰ, 60-Ⅰ, 70-Ⅰ, 80-Ⅰ, 90-Ⅰ IA 1.64 20-Ⅰ, 35-Ⅰ, 45-Ⅱ, 60-Ⅰ, 70-Ⅰ, 80-Ⅰ, 90-Ⅰ


0%

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40%

60%

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100%

1 2 3 4 5 6NC NC_M C1 C1_M C2 C2_M

Construction cost

Cost of failure

Cost of severe spalling/cracking

Cost of repair/retrofit

LCC

LCC

Deterioration risk

Deterioration risk

Maintenance Practiced

No maintenance

× (C

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C2C1NC

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31%

23%

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1%8%

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0.6%4.4%

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1.3%

28.7%

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1 2 3 4 5 6NC NC_M C1 C1_M C2 C2_M

Construction cost


Cost of failure


NC C1 C2

LCC

LCC

Deterioration risk

Deterioration risk

Maintenance Practiced

No maintenance

× (C

onst

ruct

ion

cost

)

37%

62%

1%

39.6%

60%

0.4%

42%

57.6%

0.4%

52%

0.5%

17%

30.5%

67%

5%

28%

71%

29%

1 2 30

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1 2 3 4 5 6NC NC_M C1 C1_M C2 C2_M

Construction cost



NC C1 C2

LCC

LCC

Deterioration risk

Deterioration risk

× (C

onst

ruct

ion

cost

) No maintenance

Maintenance Practiced39%

61%

42%

58%

44%

56%

63%

15%

22%

69%

1%

30%

74%

5%

21%

Fig. 12 LCC, deterioration risk and rates of each cost in the LCC of the column on the first floor.

Fig. 13 LCC, deterioration risk and rates of each cost in the LCC of the column on the sixth floor.

Fig. 14 LCC, deterioration risk and rates of each cost in the LCC of the column on the twelfth floor.


diversity of the memory cell increases with each generation and lasts until the end of the analysis. It also can be said that IA has the ability to research semi-optimal solutions (global solution and local solutions) with a high diversity in a single analysis.

6. Conclusions

In this paper, we described a system that can estimate the deterioration risk of carbonation caused by earthquakes. We also applied the immune algorithm to minimize the LCC, including the deterioration risk, and found the optimal maintenance plan for an RC beam or column. According to the results of the case study, besides the optimal maintenance plan, semi-optimal solutions with high diversity are also provided in a single analysis. In other words, the system can obtain multiple maintenance plans with high diversity, thereby allowing a user to select the suggestion he/she finds most appropriate. References AIJ (1988). “Design guideline for earthquake resistant

reinforced concrete buildings based on ultimate strength concept.” Tokyo: Architectural Institute of Japan. (in Japanese)

AIJ (2004). “Recommendations for Durability Design and Construction Practice of Reinforced Concrete.” Tokyo: Architectural Institute of Japan. (in Japanese)

Chiu, C. K., Kanematsu, M., Noguchi, T. and Nagai, H. (2007). “Optimization maintenance plan by minimizing life-cycle cost including deterioration risk due to chloride attack.” Journal of Structural and Construction Engineering (AIJ), 616, 41-47. (in Japanese)

Frangopol, D. M., Lin, K. Y. and Estes, A. C. (1997). “Life-cycle cost design of deteriorating structures.” Journal of Structure Engineering (ASCE), 123(10), 286-297.

Honma, T., Kaji, H. and Tosaka, N. (2005). “Optimization of structure system by using an immune algorithm and diversity of its solution.” Journal of Structural and Construction Engineering (AIJ), 588,

103-110. (in Japanese) JCI (1998). “Committee report of study on the

Rehabilitation of concrete structures.” Tokyo: Japan Concrete Institute. (in Japanese)

Kanda, J. and Shah, H. (1997). “Engineering role in failure cost evaluation for buildings.” Structure Safety, 19(1), 79-90.

Kanematsu, M. and Noguchi, T. (2001). “Study on optimization method of maintenance and repair scheme by applying a genetic algorithm.” Proceedings of JCI Symposium on Evaluation and Maintenance Planning for Combined Deterioration of Concrete Structures, 51-54. (in Japanese)

Kato, Y. and Uomoto, T. (2005). “Proposal for quantitative evaluation methodology of inspection value in maintenance of concrete structures based on repair-risk.” Journal of Advanced Concrete Technology, 3(3), 363-370.

Kawanishi, T., Hamasaki, H. and Masuda, Y. (2006). “Analysis of the carbonation of concrete based on investigation of existing buildings.” Journal of Structural and Construction Engineering (AIJ), 608, 9-14. (in Japanese)

Kono, M., Masuda, Y., Ochiai, R. and Karasawa, T. (2004). “Study on the carbonation suppressive effects of concrete considering the aging of coating materials for textured finishes.” Journal of Structural and Construction Engineering (AIJ), 584, 15-21. (in Japanese)

Kono, M. and Masuda, Y. (2006). “Verification of the carbonation suppressive effects evaluation method of concrete considering the aging of coating materials for textured finishes using the exposure data.” Journal of Structural and Construction Engineering (AIJ), 609, 9-14. (in Japanese)

Luh, G. C. and Chueh, C. H. (2004). “Multi-objective optimal design of truss structure with immune algorithm.” Computers & structures, 82, 829-844.

Mori, Y. and Ellingwood, B. R. (1994). “Maintaining reliability of concrete structures. II: Optimum inspection/repair.” Journal of Structure Engineering (ASCE), 120(3), 846-862.

0 200 400 600 800 1000Generations

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AH

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First floor Sixth floor

SH(IA) SH(IA)

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SAH(GA) SAH(GA)

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SAH(GA)

Fig. 15 SH, SAH of columns on the first, sixth and twelfth floors (without coating).


Nakamura, H., Kawamura, K. Onimaru, H. and Miyamoto, A. (2001). “Optimization of bridge maintenance strategies by using GA and IA techniques.” Journal of Structural Engineering, 47A, 201-210. (in Japanese)

Ozaki, T. and Morikawa, H. (2005). “Performance evaluation of RC bridges with corroded reinforcing bars based on condition rating method.” Proceedings of the Japan Concrete Institute, 27(2), 1573-1578. (in Japanese)

Stewart, M. G. and Val, D. V. (2003). “Multiple limit states and expected failure costs for deteriorating reinforced concrete bridges.” Journal of Bridge Engineering (ASCE), 8(6), 405-415.

Takahashi, T., Sakai, M., Seki, H. and Matsushima, M. (2005). “Calculation of LCC and selection system of

repairing method for reinforced concrete members exposed to sea environments.” Concrete Research and Technology (JCI), 16(3), 21-29. (in Japanese)

Tottori, S. and Miyagawa, T. (2004). “Deterioration prediction of concrete structures concerning rebar corrosion due to carbonation.” Journal of the Japan Society of Civil Engineers, 767(V-64), 35-46. (in Japanese)

Val, D. V. and Stewart, M. G. (2003). “Life-cycle cost analysis of reinforced concrete structures in marine environments.” Structure Safety, 25, 343-362.

Val, D. V. (2005). “Effect of different limit states on life-cycle cost of RC structures in corrosive environment.” Journal of Infrastructure Systems (ASCE), 11(4), 231-240.

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