TRANSITION PROBABILITY MATRIX OF BRIDGE MEMBERS DAMAGE RATING

TRANSITION PROBABILITY MATRIX OF BRIDGE MEMBERS DAMAGE RATING Hirohi Sato and Ryoji Hagiwara 2 Abtract Bridge member damage characteritic were tudied uing the inpection record. Damage can be claified into three type. It wa found that mot of the damage were claified into Type 3, where damage rating other than A wa carcely oberved. The tranition probability matrice were etimated for all the bridge member damage except for the Type 3, uing the damage rating record of the bridge up to the age of 40 year. The tranition probability wa determined o that prediction error may be minimum. The relative frequency ditribution predicted from the tranition probability matrice agreed fairly well with the inpection reult. The tranition probability p BA of Type 2, where damage rating doe not change to higher level with age, wa larger than that of Type, where damage rating change to higher level with age. The tranition probability matrice depend on the information on the repair or rehabilitation conducted to the bridge. Since the information wa not available for the data we analyzed, we cannot determine the lower left component of the matrix. The problem will be olved by analyzing tranition data obtained from two conecutive inpection reult of the ame member of the ame bridge, which wa not repaired or rehabilitated between the two inpection. Introduction There are one hundred and fifty thouand of highway bridge longer than 5 m in Japan. The highway tock have increaed in volume. Cot-effective and ytematic bridge management i required under uch ituation. Minitry of Land, Infratructure, Tranport, and Tourim iued the Periodical Inpection Manual for Bridge (Draft) [] in 2004. Now mot of the national highway bridge are being inpected according to the Manual. Data on deterioration of highway bridge member are being accumulated. On the other hand, modeling of deterioration procee have been tudied by many reearcher, and Markov proce model are ometime adopted to Bridge Management Sytem including PONTIS. In the previou paper [2], ditribution of damage rating of bridge member were tudied. A for corroion of teel main girder, and palling/ expoure of reinforcement of concrete deck, the change of their ditribution with age howed natural trend, namely Profeor, Graduate School of Sytem and Information Engineering, Univerity of Tukuba 2 Director of Planning Department, Japan Bridge Engineering Center

damage rating changed to higher level with age. But not all the damage howed thi trend. For example, damage rating did not change to higher level with age in cae of concrete deck crack. In cae of crack of teel main girder, damage rating other than A wa carcely oberved. Deterioration of ome damage of bridge member were modeled by Markov proce, and tranition probability matrice were calculated for corroion of teel main girder, palling / expoure of reinforcement of concrete deck, and concrete deck crack. In thi paper, tranition probability matrice were preented for all the bridge member damage except for damage whoe rating other than A wa carcely oberved. The relative frequency ditribution predicted from the tranition probability matrice were compared with the inpection reult. Several calculation method of tranition probability matrice were compared and dicued. Type of Damage Rating of Bridge Member According to the Periodical Inpection Manual for Bridge (Draft), damage rating of bridge member hould be given a follow: TABLE DAMAGE RATING OF BRIDGE MEMBERS [] Damage Rating State of Damage, Action Required A No damage, or the damage i o light that repair i unneceary. B Repair i neceary according to the ituation. C Prompt repair or other work i neceary. E Emergency repone i neceary to keep afety of the bridge tructure. E2 Emergency repone i neceary from the other reaon. M Maintenance work i neceary. S Detailed urvey i neceary Kind of damage to be inpected are pecified according to the member and it material [] a i hown in Table 2. In the previou paper [2], damage were claified into three type according to their deterioration characteritic: Type : Damage rating change to higher level with age. (For example, corroion of teel main girder, and palling/ expoure of reinforcement of concrete deck. ) Type 2: Damage rating doe not change to higher level with age. (For example, crack of concrete deck. ) Type 3: Damage rating other than A i carcely oberved. (For example, crack of teel main girder. ) Baed on the inpection record, damage were claified and hown in the Table 2, where

damage of Type and Type 2 are colored by yellow. It wa found that mot of the damage were claified into Type 3. The tranition probability matrice were calculated for the damage of Type and Type 2. For the damage of Type 3, tranition probability matrice were not calculated, but identity matrice eem appropriate for their tranition probability matrice. Prediction of Tranition Probability Matrix If a deterioration proce of a bridge member i aumed to be a Markov proce, and if the tranition probability matrix i aumed to be homogeneou, then the tate probability can be predicted a follow. n π( n) = π( n ) P = π(0) P () where π( n) :tate probability vector at time n π( n) = [ q ( n) q ( n) L L q ( n) ] 2 qi ( n) : probability of tate i at time n P : tranition probability matrix p p2 L L p m p2 p22 L L p2m P = M M O M M M O M p m pm2 L L pmm : tranition probability from tate i to tate j p m In cae of deterioration of bridge member, tranition data can be obtained from two conecutive inpection reult of the ame member of the ame bridge. Unfortunately, the econd inpection reult according to the Bridge Inpection Manual 2004 were not available at the time of our analyi. Therefore tranition probability wa etimated from the tate probability vector of the bridge up to the age of 40 year. The time for one-tep wa one-year. The tate probability wa aumed to be the ame a the oberved relative frequency of rating. The tranition probability wa determined o that prediction error may be minimum. The method [3] are outlined a follow. prediction error ε where y yˆ yˆ = : oberved probability of : predicted probability of i= y t-,i p T t= j= (y -yˆ ) 2 minimum tate j at time t tate j at time t (2) (3)

on the condition that p kh j= = 0 p =, (k,h) J i =, 2, L where P : tranition probability matrix y t :tate probability vector at time t Applying the method of Lagrange undetermined multiplier, T 2 [ (y- yt-,i p ) + λ i( p- ) + μkh pkh ] = 0 p T y t= j= i= T y - ( y t-,i t= k= t= t-,i y t-, k ) p kj i= j= = λ Equation (7) can be rewritten into Y - ZP = λξ where [ Y ] [ Z ] T + T t= [ λ ] i λ i [ U ] μ [ξ] i U y T t= t-,i y y t-,i t-,j y i + μ ( μ (k,h) J = 0 when (i, j) J) (4) (5) (6) (7) (8) When all of P = Z - Y p kh 0, then all of μ kh = 0, which mean U = 0. Therefore, (9) In the calculation, Equation (9) wa ued at firt. A i clear from Equation (6), the calculated tranition probability matrice automatically atify the Equation (5), however, the calculated tranition probability doe not necearily take value between 0 and. When negative value wa obtained, the correponding tranition probability wa aumed to be 0. Then the tranition probability matrice were calculated again a follow [3]. μ wa calculated from the following equation. - P = Z (Y + Uξξ = 0 (k,h) J p kh T - U) Then, P can be calculated by ubtituting the μ into the equation (0). (0)

Uing the above method [Method ], the tranition probability matrice were etimated for damage of Type and Type 2. The etimated tranition probability matrice are hown in the Table 3. The relative frequency ditribution predicted from the tranition probability matrice are hown in the Figure.-.3. for corroion of teel main girder, palling/ expoure of reinforcement of concrete deck, and crack of concrete deck. Since the frequencie other than A, B and C were very few, only the three tate were conidered in the calculation. The figure at the top how the inpection reult. The figure in the middle how the relative frequency ditribution predicted from the following equation [Prediction ]. πˆ ( n) = π( n ) P ( ) where, πˆ ( n): predicted tate probability vector at time n. The figure at the bottom how the relative frequency ditribution predicted from the following equation [Prediction 2]. n π ˆ ( n) = π(0) P ( ) The prediction error defined a in Equation (2) are alo hown in the figure. The predicted relative frequency ditribution agree fairly well with the inpection reult. Dicuion Corroion of teel main girder and palling/ expoure of reinforcement of concrete deck belong to the damage of Type, where damage rating change to higher level with age. On the other hand, crack of concrete deck belong to the damage of Type 2, where damage rating doe not change to higher level with age. The tranition probability p AA of Type i larger than that of Type 2, however, the tranition probability p BA of Type 2 i larger than that of Type. Tranition probabilitie in the lower left of the tranition probability matrix, p BA for example, how the effect of repair or rehabilitation. If no repair or rehabilitation work i done, lower left component of the matrix hould be equal to 0. The information on the repair or rehabilitation i not available for thee data. If it i aumed that repair or rehabilitation were conducted, and that thee effect were accurately reflected in thee tranition probabilitie, then tranition probability matrix in cae of no repair or rehabilitation can be obtained a in the Table 4.2. TABLE 4. ORIGINAL TRANSITION PROBABILITY MATRIX A B C A p AA p AB p AC B p BA p BB p BC C p CA p CB p CC

TABLE 4.2 MODIFIED TRANSITION PROBABILITY MATRIX IN CASE OF NO REPAIR OR REHABILITATION A B C A p AA p AB p AC B 0 p BB /(p BB +p BC ) p BC /(p BB +p BC ) C 0 0 If it i aumed that no repair or rehabilitation wa conducted, tranition probability matrice can be calculated applying the Equation (4) to the lower left component of the matrix [Method 2]. The reult i hown in the Figure 2. for the corroion of teel main girder. p AA or p BB in the matrix of the Figure 2. i larger than thoe in the Figure. where it wa aumed that repair or rehabilitation wa conducted. The error hown in the middle figure are not o different between the Figure. and the Figure 2., however, the error hown in the bottom figure of the Figure 2. i much larger than that of Figure.. In the prediction of the tranition probability matrice in the Figure. and the Figure 2., the following prediction error for the middle figure were minimized. prediction error ε T 2 (y - yt-,i p ) t= j= i= On the other hand, prediction error for the bottom figure are a follow. prediction error ε 2 where, yˆ yˆ 0,i t,i = y = k = 0,i yˆ t-,k t= j= i= p T ki (y - t yˆ t-,i By minimizing the error ε 2, the third tranition probability matrix wa predicted [Method 3], and the reult are hown in the Figure 2.2. In the prediction, p AC a well a lower left component were aumed to be 0. Although the error in the middle figure wa lightly larger than thoe in the Figure. and 2., the error in the bottom wa much maller a wa expected. Since the information on the repair or rehabilitation i not available for thee data, we cannot conclude the lower left component of the matrix. The problem will be olved by analyzing tranition data obtained from two conecutive inpection reult of the ame member of the ame bridge, which wa not repaired or rehabilitated between the two inpection. Concluion Bridge member damage characteritic were tudied uing the inpection record. Some of the finding are a follow. p ) 2

. Damage can be claified into three type. It wa found that mot of the damage were claified into Type 3, where damage rating other than A wa carcely oberved. 2. The tranition probability matrice were etimated for all the bridge member damage except for the Type 3, uing the damage rating record of the bridge up to the age of 40 year. The tranition probability wa determined o that prediction error may be minimum. The relative frequency ditribution predicted from the tranition probability matrice agreed fairly well with the inpection reult. The tranition probability p AA of Type wa larger than that of Type 2, however, the tranition probability p BA of Type 2 wa larger than that of Type. 3. The tranition probability matrice depend on the information on the repair or rehabilitation conducted to the bridge. Since the information wa not available for the data we analyzed, we cannot determine the lower left component of the matrix. The problem will be olved by analyzing tranition data obtained from two conecutive inpection reult of the ame member of the ame bridge, which wa not repaired or rehabilitated between the two inpection. Reference ) National Highway and Rik Management Diviion, Road Bureau, MLITT: Periodical Inpection Manual for Bridge (Draft), 2004, 3, (in Japanee) 2) H. Sato: Ditribution of damage rating of bridge member and a few conideration, 24th US - Japan Bridge Engineering Workhop, 2008, 9 3) H. Morimura and Y. Takahahi: Markov Analyi, JUSE Pre, Ltd, 979, pp.287-29 (in Japanee)

Member Damage Member Damage TABLE 2 BRIDGE MEMBERS AND THEIR DAMAGES Steel Steel Steel Main Steel Cro Steel Steel Plate Steel Abutment or Expanion Girder Beam Stringer Deck Bearing Pier Joint Corroion Corroion Corroion Corroion Corroion Corroion Corroion Crack Crack Crack Crack Crack Crack Crack Looene or Looene or Looene or Looene or Looene or Looene or Looene or Fracture Fracture Fracture Fracture Fracture Fracture Fracture Paint Failure Paint Failure Paint Failure Paint Failure Paint Failure Paint Failure Paint Failure Gap Gap Gap Function Failue Gap Water Surface Sound or Sound or Sound or Sound or Sound or Leakage or Roughne Vibration Vibration Vibration Vibration Vibration Ponding Dirt and Debri Sag, Move or Slope Concrete Concrete Concrete Concrete Concrete Other Abutment or Main Girder Cro Beam Stringer Deck Bearing Pier Crack Crack Crack Crack Crack Fracture Spalling or Spalling or Expoure of Expoure of Reignforcement Reignforcement Water Leakage or Efflorecence Damage of Water Leakage or Efflorecence Spalling or Spalling or Expoure of Expoure of Reignforcement Reignforcement Water Leakage or Efflorecence Damage of Water Leakage or Efflorecence Spalling or Expoure of Reignforcement Water Leakage or Efflorecence Damage of Reignforcement Reignforcement Damage of Reignforcement Reignforcement Damage of Reignforcement Delamination Delamination Delamination Delamination Delamination Gap Gap Gap Gap Anchor Anchor Anchor Anchor Anchor Problem Problem Problem Problem Problem Change of Change of Change of Change of Change of Colour Colour Colour Colour Colour Water Water Water Water Water Leakage or Leakage or Leakage or Leakage or Leakage or Ponding Ponding Ponding Ponding Ponding Sound or Vibration Sound or Vibration Sound or Vibration Sound or Vibration Sound or Vibration Function Failue Change of Colour Water Leakage or Ponding Dirt and Debri Dirt and Debri Other Expanion Joint Gap Surface Roughne Change of Colour Water Leakage or Ponding Sound or Vibration Dirt and Debri Legend other : Tranition Probability Matrice were predicted. : Tranition Probability Matrice were not predicted becaue damage rating other than A were carcely oberved.

TABLE 3 TRANSITION PROBABILITY MATRICES OF BRIDGE MEMBERS DAMAGE RATING Steel Main Girder, Corroion Steel Cro Beam, Corroion Steel Stringer, Corroion Steel Plate Deck, Corroion 0.925 0.075 0.000 0.955 0.038 0.007 0.904 0.078 0.08 0.858 0.5 0.027 0.220 0.553 0.227 0.265 0.54 0.95 0.465 0.372 0.63 0.242 0.542 0.26 0.000 0.949 0.05 0.000.000 0.000 0.25 0.62 0.37 0.07 0.80 0.28 Steel Abutment or Pier, Corroion Steel Bearing, Corroion Steel Expanion Joint, Corroion 0.836 0.6 0.047 0.899 0.0 0.000 0.904 0.096 0.000 0.520 0.384 0.096 0.226 0.542 0.23 0.505 0.420 0.076 0.434 0.463 0.03 0.000 0.660 0.340 0.000 0.807 0.93 Steel Main Girder, Paint Failure Steel Cro Beam, Paint Failure Steel Stringer, Paint Failure Steel Plate Deck, Paint Failure 0.707 0.277 0.07 0.767 0.204 0.029 0.588 0.356 0.056 0.833 0.58 0.009 0.323 0.602 0.075 0.400 0.520 0.080 0.672 0.239 0.089 0.236 0.647 0.8 0.000 0.824 0.76 0.9 0.602 0.208 0.46 0.54 0.340 0.000 0.8 0.89 Steel Abutment or Pier, Paint Failure Steel Bearing, Paint Failure Steel Expanion Joint, Paint Failure 0.646 0.322 0.032 0.79 0.209 0.000 0.936 0.064 0.000 0.655 0.27 0.074 0.339 0.504 0.57 0.504 0.445 0.050 0.632 0.253 0.5 0.000 0.932 0.068 0.000.000 0.000 Concrete Main Girder, Crack Concrete Cro Beam, Crack Concrete Deck, Crack Concrete Abutment or Pier, Crack 0.966 0.03 0.004 0.920 0.077 0.003 0.658 0.337 0.005 0.604 0.384 0.02 0.569 0.43 0.000 0.694 0.306 0.000 0.439 0.542 0.09 0.864 0.36 0.000.000 0.000 0.000.000 0.000 0.000 0.02 0.444 0.535.000 0.000 0.000 Concrete Main Girder, Spalling or Expoure of Reignforcement Concrete Cro Beam, Spalling or Expoure of Reignforcement Concrete Deck, Spalling or Expoure of Reignforcement Concrete Abutment or Pier, Spalling or Expoure of Reignforcement 0.973 0.026 0.002 0.97 0.029 0.000 0.98 0.03 0.006 0.99 0.009 0.000 0.65 0.25 0.34 0.330 0.535 0.34 0.000 0.92 0.088 0.053 0.808 0.39 0.000 0.94 0.059 0.000 0.6 0.389 0.030 0.295 0.675 0.000 0.986 0.04 Concrete Cro Beam, Water Concrete Deck, Water Leakage Concrete Abutment or Pier, Leakage or Efflorecence or Efflorecence Water Leakage or Efflorecence 0.89 0.75 0.006 0.693 0.307 0.000 0.957 0.043 0.000 0.87 0.83 0.000 0.26 0.738 0.046 0.263 0.72 0.025 0.704 0.228 0.067 0.53 0.537 0.309 0.000.000 0.000 Concrete Deck, Delamination Other Expanion Joint, Water Concrete Abutment or Pier, Leakage or Ponding Water Leakage or Ponding 0.99 0.009 0.000 0.846 0.094 0.06 0.954 0.045 0.00 0.05 0.648 0.246.000 0.000 0.000 0.420 0.549 0.030 0.000 0.297 0.703 0.966 0.034 0.000 0.48 0.376 0.43 Other Bearing, Change of Colour Other Expanion Joint, Change Other Expanion Joint, of Colour 0.978 0.022 0.000 0.946 0.038 0.06 0.965 0.025 0.00 0.438 0.545 0.07 0.57 0.427 0.002 0.530 0.358 0..000 0.000 0.000 0.070 0.569 0.36 0.575 0.329 0.097 Legend: Member, Damage P AA P AB P AC P BA P BB P BC P CA P CB P CC A:No damage, or the damage i o light that repair i unneceary. B:Repair i neceary according to the ituation. C:Prompt repair or other work i neceary.

CORROSION OF STEEL MAIN GIRDER(INSPECTION) RELATIVE FREQUENCY OF 0 PC PB PA CORROSION OF STEEL MAIN GIRDER(PREDICTION, ERROR=0.567) RELATIVE FREQUENCY OF 0 PCp PBp PAp CORROSION OF STEEL MAIN GIRDER(PREDICTION2, ERROR=0.852) RELATIVE FREQUENCY OF 0 PCp2 PBp2 PAp2 FIG.. DISTRIBUTION OF DAMAGE RATING FOR CORROSION OF STEEL MAIN GIRDERS (TOP: INSPECTION (SOURCE: MINISTRY OF LAND, INFRASTRUCTURE, TRANSPORT, AND TOURISM), MIDDLE: PREDICTION, BOTTOM: PREDICTION2, RIGHT: TRANSITION PROBABILITY MATRIX OBTAINED FROM METHOD) A B C A 0.925 0.075 0.000 B 0.220 0.553 0.227 C 0.000 0.949 0.05

SPALLING OR EXPOSURE OF REIGNFORCEMENT OF CONCRETE DECK (INSPECTION) RELATIVE FREQUENCY OF 0 PC PB PA SPALLING OR EXPOSURE OF REIGNFORCEMENT OF CONCRETE DECK (PREDICTION, ERROR=0.34) RELATIVE FREQUENCY OF 0 PCp PBp PAp SPALLING OR EXPOSURE OF REIGNFORCEMENT OF CONCRETE DECK (PREDICTION2, ERROR=0.633) RELATIVE FREQUENCY OF 0 PCp2 PBp2 PAp2 FIG..2 DISTRIBUTION OF DAMAGE RATING FOR SPALLING OR EXPOSURE OF REIGNFORCEMENT OF CONCRETE DECK (TOP: INSPECTION (SOURCE: MINISTRY OF LAND, INFRASTRUCTURE, TRANSPORT, AND TOURISM), MIDDLE: PREDICTION, BOTTOM: PREDICTION2, RIGHT: TRANSITION PROBABILITY MATRIX OBTAINED FROM METHOD) A B C A 0.98 0.03 0.006 B 0.000 0.92 0.088 C 0.030 0.295 0.675

CRACK OF CONCRETE DECK(INSPECTION) 0 PC PB PA CRACK OF CONCRETE DECK(PREDICTION, ERROR=0.428) 0 PCp PBp PAp CRACK OF CONCRETE DECK(PREDICTION2, ERROR=0.48) 0 PCp2 PBp2 PAp2 FIG..3 DISTRIBUTION OF DAMAGE RATING FOR CRACK OF CONCRETE DECK (TOP: INSPECTION (SOURCE: MINISTRY OF LAND, INFRASTRUCTURE, TRANSPORT, AND TOURISM), MIDDLE: PREDICTION, BOTTOM: PREDICTION2, RIGHT: TRANSITION PROBABILITY MATRIX OBTAINED FROM METHOD) A B C A 0.658 0.337 0.005 B 0.439 0.542 0.09 C 0.02 0.444 0.535

CORROSION OF STEEL MAIN GIRDER(INSPECTION) 0 PC PB PA CORROSION OF STEEL MAIN GIRDER(PREDICTION, ERROR=0.68) 0 PCp PBp PAp CORROSION OF STEEL MAIN GIRDER(PREDICTION2, ERROR=2.77) 0 PCp2 PBp2 PAp2 FIG. 2. DISTRIBUTION OF DAMAGE RATING FOR CORROSION OF STEEL MAIN GIRDERS (TOP: INSPECTION (SOURCE: MINISTRY OF LAND, INFRASTRUCTURE, TRANSPORT, AND TOURISM), MIDDLE: PREDICTION, BOTTOM: PREDICTION2, RIGHT: TRANSITION PROBABILITY MATRIX OBTAINED FROM METHOD2) A B C A 0.975 0.025 0.000 B 0.000 0.946 0.054 C 0.000 0.000.000

CORROSION OF STEEL MAIN GIRDER(INSPECTION) 0 PC PB PA CORROSION OF STEEL MAIN GIRDER(PREDICTION, ERROR=0.687) 0 PCp PBp PAp CORROSION OF STEEL MAIN GIRDER(PREDICTION2, ERROR=0.357) 0 PCp2 PBp2 PAp2 FIG. 2.2 DISTRIBUTION OF DAMAGE RATING FOR CORROSION OF STEEL MAIN GIRDERS (TOP: INSPECTION (SOURCE: MINISTRY OF LAND, INFRASTRUCTURE, TRANSPORT, AND TOURISM), MIDDLE: PREDICTION, BOTTOM: PREDICTION2, RIGHT: TRANSITION PROBABILITY MATRIX OBTAINED FROM METHOD3) A B C A 0.985 0.05 0.000 B 0.000 0.985 0.05 C 0.000 0.000.000