NUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED CONCRETE STRUCTURES WITH CRACKS

VIII Internatinal Cnference n Fracture Mechanics f Cnete and Cnete Structures FraMCS-8 J.G.M. Van Mier, G. Ruiz, C. Andrade, R.C. Yu and X.X. Zhang (Eds) NUMERICAL SIMULATION OF CHLORIDE DIFFUSION IN REINFORCED CONCRETE STRUCTURES WITH CRACKS CHRISTOPHER K Y LEUNG * DONGWEI HOU *, * Department f Civil and Envirnmental Engineering The Hng Kng University f Science and Technlgy (HKUST) Clear Water Bay, Kwln, Hng Kng e-mail:hudw@ust.hk, www.ust.hk Key wrds: Chlride diffusin; effect f acking; equivalent diffusin cefficient; durability Abstract: Steel crrsin is a majr prblem with cnete structures. Fr cnete structures expsed t chlrides, nce a itical chlride cncentratin is reached at the steel surface, crrsin starts t ccur. Fr a given cver thickness, the crrsin initiatin time depends n the chlride diffusivity. The diffusivity fr cnete is nrmally measured at the un-acked state. Hwever, in reinfrced cnete design, cnete members are allwed t ack in tensin. Once acking ccurs, chlride can als diffuse thrugh the ack at a faster rate gverned by the ack diffusivity D. In the present study, finite element analysis is perfrmed t study the effect f acking n crrsin initiatin in cnete with different values f. Fr varius ack width and cver thickness, the time fr itical chlride cncentratin t be reached at steel intersected by the ack is determined. Based n the simulatin results, we derive the empirical equatin fr an equivalent diffusin cefficient (D eq ) which can be emplyed directly in the slutin f the ne-dimensinal diffusin equatin t calculate the crrsin initiatin time. The study prvides a cnvenient but accurate apprach fr predicting crrsin initiatin under real wrld situatins where acks are present in a cnete structure. INTRODUCTION Steel crrsin is a majr prblem with cnete structures. In the alkaline envirnment inside cnete, steel is in the passivated state with a thin surface layer f stable irn xide t prtect it frm crrsin. Fr cnete structures expsed t chlrides, the chlride cncentratin at the surface f steel reinfrcements will inease with time due t the diffusin prcess. Once a itical chlride cncentratin is reached, the steel is depassivated and crrsin starts t ccur. Fr a given cver thickness, the crrsin initiatin time depends n the chlride diffusivity f the cnete. Many researchers have measured the diffusivity f cnete at its virgin state [-3].

Hwever, in reinfrced cnete design, cnete members are allwed t ack in tensin. The effect f acks n chlride diffusivity is therefre an imprtant issue t be studied. In Aldea et al. [4], the Rapid Chlride Penetratin Test (RCPT) was perfrmed n laded cnete discs with disete tensile acks frm.5mm t.4mm in width. Fr nrmal strength cnete samples, chlride permeability was fund t be much less sensitive t ack pening than high strength cnete with lwer permeability in the un-acked state. In Gerard and Marchand [5], the cefficient f chlride diffusivity (D Cl ) was measured fr specimens acked under freezing/thawing with the tw cell migratin test, in which a cnete disc is placed between tw cells with slutins f different chlride cncentratin. Results indicated that D Cl wuld inease with the number and pening f acks. Olga and Htn [6] perfrmed bulk chlride diffusin test n cnete specimens cntaining artificial acks with penings between. mm t.68 mm. In the test, ne face f the specimens was expsed t chlrides, and D Cl was determined frm the chlride prfile in the specimen. Accrding t the test results, the value f D Cl is nt dependent n ack pening within the tested range. Hwever, since the acks in the samples are artificial nes intrduced in a certain special way, the findings may nt apply quantitatively t cnete with acks frmed under lading. T date, the mst cmprehensive study n the effect f acks n chlride diffusivity was cnducted by Djerbi et al [7]. In this wrk, cnete discs made with three different kinds f cnete were subjected t split tensin t prduce disete acks f varius penings. The acked discs (as well as cntrl specimens with n acks) were tested in the chlride migratin cell with bth chlride cncentratin gradient and electric ptential ass the tw sides f the sample. The value f D Cl fr each specimen was btained frm the change f chlride cncentratin in the dwnstream cell after steady state had been reached. Based n a parallel flw mdel first prpsed by Gerard and Marchand [5] fr the steady state, the chlride diffusivity f the ack (D ) was determined and pltted against the ack width. Interestingly, D was fund t be dependent n ack width but nt the diffusivity f un-acked cnete ( ). This explains the bservatin in Aldea et al [4] that a ack has larger effect n cnete with lwer diffusivity, as D is much higher than fr such a case. Knwing the value f D fr a particular ack width, analysis f the chlride diffusin prcess in acked cnete can take int cnsideratin the ineased diffusivity alng the ack. In this study, the finite element methd is emplyed t study the effect f ack width n crrsin initiatin in cnete with varius values f. In the analysis, adjacent acks are assumed t be sufficiently far away frm ne anther s we can neglect interactive effects and fcus n a single ack. Based n the simulatin results, an empirical equatin will be derived fr the equivalent diffusin cefficient (D eq ) which can be used directly fr calculating the crrsin initiatin time in acked cnete. NUMERICAL SIMULATION. Outline f the simulatin apprach Fr cnete structures expsed t the chlride envirnment, chlride diffusin thrugh the cver layer can ften be cnsidered as a -D diffusin prcess, Fig. illustrates the typical situatin f a acked cnete member with ne f its surfaces expsed t chlride. If the ack is nt present, and the chlride cncentratin is cnstant alng the surface f the member (in the y-directin), -D diffusin ccurs in the x-directin. An analytical slutin

fr the chlride cncentratin at any pint within the cnete can then be btained. Fr cnete structures cntaining acks, the rate f chlride diffusin thrugh acks is much higher than that thrugh cnete matrix. The resulted cncentratin gradient between the ack and adjacent cnete will induce lateral diffusin f chlride alng the y-directin as well. The diffusin prblem then becmes -D in nature. Under such a situatin, an analytical slutin is very difficult t derive s the finite element methd is emplyed t determine the chlride cncentratin within the member at varius times. The time fr itical chlride cncentratin t be reached at a given cnete cver can then be btained. In this study, adjacent acks in the cnete structure are assumed t be sufficiently far away frm ne anther, s there is n interactive effect amng the acks. It is then nly necessary t cnsider a single ack in the finite element mdel. T simulate the diffusin prcess, the ANSYS. FEA sftware is emplyed. Fig. shws a typical finite element mesh f the diffusin dmain. The dimensin f the mdel is mm (alng the surface) times mm (alng the directin f the ack). The fcus f the wrk is t determine the chlride cncentratin alng the ack, because steel crrsin will ccurs first at the sectin intersected by the ack. The chsen length f the mdel is mm, which is significantly higher than the typical steel cver f 3 t mm. When itical chlride cncentratin is reached at the steel, the chlride cncentratin at the lwer bundary is fund (in the simulatin results) t be very small s the bundary has little effect n the diffusin prcess. Similarly, with the width f mm, the calculated chlride cncentratins at the vertical bundaries were fund t be much lwer than that alng the ack (at the same value f x), indicating that the bundaries in the mdel are sufficiently far away frm the ack. T btain accurate numerical results, the mesh size f cnete matrix is kept within mm, while the mesh size f the ack is refined t less than.5 mm. The transitin frm small t large elements is shwn in Fig.. The maximum permitted time step is taken t be ITS δ / 4D, where δ is the mesh size at the pint with highest cncentratin gradient (alng the upper bundary in this case), and D is the crrespnding diffusin cefficient. Fig.. Diagram f chlride diffusin in acked cnete Fig. Mesh f matrix and ack in FEM While the range f fr cnete is well dcumented, D is dependent n the ack width (w). Based n experimental measurements, Djerbi et al [7] prpsed empirical equatins t relate D t w. When w ineases frm 3 t 8 μm, D ineases frm - t - m /s. Fr larger values f w, D remains cnstant at - m /s. In the finite element simulatin, instead f fllwing this relatin directly, the value f D fr a given w is allwed t vary within a certain 3

range. This is t accmmdate the pssibility f different D vs w relatin that may be revealed frm future experiments. Specifically, D varies frm - t - m /s. Als, is taken t range frm. - m /s t - m /s while w ranges frm 3μm t μm. Simulatins with varius cmbinatins f parameters D, and w are perfrmed t cver the whle range. Fr cnvenience, the chlride cncentratin n the member surface is fixed at C = g/m 3. The actual value f C is nt imprtant as it is the C/C rati which is f interest. The calculated diffusin time is up t years.. Simulatin results Typical simulatin results are shwn in Fig.3 fr w=6 μm and diffusin time f ne year. D is taken t be 8 - m /s. In the simulatin, we are cmparing the real situatin (i.e., a dmain with a ack) t tw limiting cases: case I with diffusin thrugh un-acked cnete with D= =.5 - m /s and case II with -D diffusin thrugh the ack alne, with D=D =8 - m /s. Fig.3(a) shws the chlride cncentratin fr the three cases: Case I n the left, the actual situatin fr acked cnete in the middle and Case II n the right. It is bvius that the diffusin prcess is fastest when chlride is assumed t diffuse thrugh the ack alne, with n lateral flw. In reality, hwever, due t the cncentratin gradient in the y-directin, chlride penetratin in the x-directin can be significantly slwed dwn, as bserved frm the results fr acked cnete and Case II. Als, accrding t the result fr acked cnete, the ack has little influence n diffusin at lcatins far away frm the ack. At such lcatins, the diffusin prcess can be cnsidered as -D and gverned by the surface cncentratin alne. Fig. 3(b) shws the variatin f nrmalized chlride cncentratin with distance frm surface (x) fr Case I (unacked) and variatin (a) (b) C/C.9.8.7.6.5.4.3.. Diffusin with D Diffusin in ack Diffusin with D C/C -x (ne year) w = 6 m D =8 - m /s =.5 - m /s 4 6 8 4 6 8 x(cm) Fig. 3 Chlride diffusin in acked cnete cmpared t limiting cases. (a) Plt f chlride cncentratin, (b) Variatin f chlride cncentratin with distance frm surface. alng the ack in Cases II and III. The results again illustrate that it is imprtant t cnsider the -D nature f the diffusin prblem. Specifically, if ne des nt accunt fr the lateral diffusin (i.e., the diffluence effect), chlride penetratin is severely ver-estimated as indicated by the result fr Case II. Based n physical cnsideratins, ne can argue that the diffluence effect deeases with the rati D /, but ineases with the rati A l /A w, where A l is the area fr lateral diffusin t ccur ass the ack surfaces, and A w is the area fr direct diffusin thrugh the ack. A l /A w is prprtinal t the rati x/w. As w is ften very small, x/w can be very large. Therefre, even thugh D is much higher than in many cases, the diffluence effect can still be significant. Frm this analysis, we can 4

deduce that the chlride diffusin thrugh ack in high strength cnete will be faster than that in rdinary cnete when the tw kinds f cnete have the same width f acks (i.e. the same D ), because the high strength cnete wns the lwer matrix diffusin cefficient and thus induces the weaker diffluence effect n the chlride diffusin thrugh ack than rdinary strength cnete. (a) w = 6 μm; D =8 - m /s; =. - m /s. (b) w = 5μm; D = - m /s =. - m /s. Fig.4 shws three types f chlride distributin in acked cnete with different cmbinatins f D, and w after diffusin has ccurred fr 5 years. When the D / is high and/r w is large, diffusin thrugh the ack will be much faster than that thrugh the cnete matrix, as shwn in Fig.4 (a) and Fig.4 (b). When D / is lw and/r w is small, as in Fig.4(c), the effect f the ack is nt t significant. In practice, we are mst interested in the chlride penetratin alng the ack, as steel crrsin will ccur first at the ack s vicinity. T avid the need t perfrm cmputatinal analysis fr every single case, an empirical apprach fr analyzing chlride diffusin thrugh the ack is prpsed. One-dimensinal diffusin is assumed t ccur and the equivalent diffusin cefficient (D eq ) is determined empirically frm the fitting f numerical results. With a general expressin fr D eq, the chlride cncentratin alng the ack under any cmbinatin f D,, w and time can be btained, details f the empirical apprach are discussed in the fllwing sectins. 3 EMPIRICAL MODELING OF DIFFUSION THROUGH CRACK (c) w = 6μm; D =8 - m /s = - m /s 3. Equivalent diffusin cefficient As mentined abve, diffusin thrugh the ack is affected by the diffluence effect in the y-directin (which is perpendicular t the ack). Based n mass cnservatin, the chlride cncentratin (C ) alng the ack can be btained frm the fllwing equatin: C t D C x D w G( x) () Fig. 4 Typical simulatin results f chlride diffusins in acked cnetes where G(x) is abslute value f C/ y y=+w/ (i.e., the bundaries f the ack), in at 5

unit f g/m 4. As eqn () is very difficult t slve, a semi-empirical apprach is prpsed t replace it with the equatin fr -D diffusin: Where D eq C t D D eq D C x G(x) C w x Fr the sake f simplicity, D eq can be further expressed in the fllwing frm D e q () (3) D - K D (4) Eqn (4) indicates that D eq will be smaller than D due t the diffluence effect. It is clear that D eq and K are nt cnstants. Generally speaking, they are functins f varius parameters including D (m /s), (m /s) and w (m), as well as distance frm the external bundary x (m) and diffusin age t (s). In Djerbi et al [7], a relatin between D and w was prpsed. Hwever, as mentined abve, different researchers have reprted different effects f w n D. In the present study, t make the mdel mre general and applicable t data frm different investigatins (including thse t be btained in the future), D and w are treated as tw separate variables. T btain D eq fr a particular cmbinatin f D,, w, x and t frm the finite element result, the classical slutin fr ne-dimensin diffusin is emplyed: x C (x, t) Cerfc (5) De qt Eqn (5) is nt a slutin t eqn () because D eq is nt cnstant. The use f this equatin t btain D eq is simply fr cnvenience. Once we have carried ut a sufficient number f numerical simulatins, an empirical expressin fr D eq will be determined in terms f the varius parameters thrugh data fitting. With this empirical expressin, the chlride cncentratin alng the ack can be easily btained frm eqn (5). 3. Range f valid numerical data As in any cmputatinal methd, results f the finite element mdel are subject t numerical errr. In the fitting f finite element results, it is imprtant t avid data with high inaccuracy. The tw majr reasns fr inaccurate numerical data are discussed belw. In the derivatin f the diffusin equatin, the transfer speed f chlrides is assumed t be infinite [8]. As a result, the chlride cncentratin at any pint in the cmputatinal dmain is nn-zer nce the diffusin prcess begins (i.e.t > ). This is clearly revealed frm eqn (5), which is the slutin fr the -D diffusin equatin. At lcatins far away frm the surface, very small values f chlride cncentratin will be calculated. Fr such results, the numerical errr will be very high, leading t inaccurate values f D eq in eqn (5). Fr a given distance frm the surface, the calculated cncentratin can nly be cnsidered valid after the diffusin prcess has gne n fr a sufficiently lng time. Determinatin f this minimum time value is best illustrated with the un-acked case, the results f which are given in Fig.5. Fig.5(a) shws the variatin f matrix diffusin cefficient (calculated frm numerical results accrding t eqn(5)) with distance frm surface x. Fr diffusin ages f 6 mnths, year, 3 years, 5 years and years, begins t deviate frm the crrect value f. - m /s beynd the distances f. cm,.8 cm, 3.cm, 4. cm, 5.cm and 5.5 cm respectively.. The inease f is caused by the significant relative errr f the lw cmputed chlride cncentratin. Fr a certain diffusin age, the finite element result can nly 6

be cnsidered valid fr pints within a certain distance frm the surface. Beynd the itical distance, ntable errr arises. Cnversely, fr a certain distance frm the surface, nly after a sufficient lng time f diffusin is the simulated chlride cncentratin valid and hence suitable fr cmputing. T help identify the itical distance fr data t be valid, the derivative f ( / x) is pltted against x fr different diffusin times in Fig. 5(b). The derivative is initially zer but begins rising after a certain distance frm the surface is reached. The itical pint (i.e. the pint f abrupt inease) is cnsistent with the demarcatin pint between valid and invalid data n the -x curve in Fig. 5(a), but it can be picked up mre easily n the / x-x curve. Fr the case with a ack, a similar apprach t determine the range f valid data can be adpted. Curves f D eq / x vs x at different times fr diffusin thrugh a ack are shwn in Fig. 6. The derivative f D eq shuld apprach zer with ineasing distance x, but an abrupt inease can be bserved due t errrs in the numerical results. The itical distance (fr a particular time) can again be picked up n the curve f D eq / x vs x. A secnd surce f errr is arising frm reflectin effect f the bttm bundary f the finite element mdel. This effect becmes significant fr high ack diffusin cefficient D and large ack width w. T examine this effect, results btained frm ur standard finite element mdel with depth f mm is cmpared t that frm a mdel f 4mm depth fr the same values f D,, and w. Results up t a distance f mm are shwn in Fig.7. D eq / x vsx curves fr the 4mm case (brken lines) shw an expected deeasing trend as there is little effect frm the far away bundary. Fr the mm case (slid lines), the reflectin effect f bttm bundary causes the derivative f D eq t inease. The lnger the diffusin age, the higher is the cncentratin near the bttm bundary and hence mre significant is the reflectin effect. A secnd itical distance is btained frm the rising part f the curve when distance is mm. Fr a given cmbinatin f D,, w, the tw itical distances desibed abve can be btained at a given time. The smaller f these tw distances is then pltted against time, as illustrated in Fig.8 t identify the range f valid data. Fr instance, at a distance 5 cm frm the surface, the valid data shuld be selected between time t and t in Fig.8. Befre t, the numerical results suffer frm significant relative errr, while after t, the results are affected by bundary reflectin. In the fllwing sectin, nly results in the valid range are used as data fr the fitting f K fr determining an empirical expressin fr D eq. (a) (b) ( - m /s) /x ( - m/s) 8 7 6 5 4 3 9 8 7 6 5 4 3 D O -x =. - m /s w=3 m.5,,3,5,8, (year) 4 6 8 4 6 8.5,,3,5,8, (year) =. - m /s w=3 m 4 6 8 4 6 8 Fig. 5 Validity f calculated by simulatin results (a)variatins f with x ; (b) Variatins f derivative f t x with x 7

D eq /x ( - m/s) D eq /x ( - m/s) 9 8 7 6 5 4 3.5,,3,5,8, (year) D = - m /s =. - m /s w=3 m 4 6 8 4 6 8 Fig. 6 Curves f D eq / x vs x calculated frm simulatin results 9 8 7 6 5 4 3 D = - m /s =.5 - m /s w=5 m.5,,3,5,8, (year) 4 6 8 4 6 8 Fig. 7 Effect f bttm bundary reflectin n curves f x (m)..8.6.4...8.6.4. Fig. 8 t (x=5cm) D eq / x vs x t (x=5cm) Valid Curve x-t D =8 - m /s =. - m /s w=6 m valit pint 36 7,8,44,8,6,5,88 3,4 3,6 T (d) Range f valid numerical results fr fitting 3.3 Mdeling and data fitting As discussed abve, the factr K in eqn (4) is gverned by D (m /s), (m /s), w (m), x (m) and t (s). T btain an empirical relatin between K and the varius parameters, dimensinal analysis is first carried ut t see hw the parameters can be cmbined. The dimensin f K is unity. Dimensins f parameters f D,, x, w, t are M T -, M T -, M, M and T respectively. Accrding t Buckingham π thery [9], the fllwing equatin can be btained: - a - a a3 a4 a5 (M T ) (M T ) M M T (6) The relatins between the indices are then given by: a - a a a3 a - a a 5 4 (7) Three pssible slutins t eqn (7) are (a, a, a 3, a 4, a 5 )=(,,-,,), (,-,,,) and (,,,-,). Based n these three slutins, the independent parameters can be cmbined as t/x, D / and x/w. With the dimensinal analysis methd, the number f independent parameters is reduced frm 5 t 3. In additin, the relatinship in terms f dimensinless parameters can better reflect the physical nature f the diffusin behavir. Frm the numerical results, D eq is first calculated and K is btained frm eqn(4). We then attempt t relate K t t/x and the fllwing functin is fund t prvide a very gd fit. D K D Dt k x - b Dt k x b (8) The first term in eqn (8) indicates that when D is similar t, K is very small, s D eq is similar t D (r ). In ther wrds, the ack has little effect n the diffusin prcess. Substituting eqn (8) int eqn (4), D eq can be expressed as D eq Dt D Dk x b (9) Dt k x b 8

The parameters k and b are then expressed as functins f D / and x/w, and the fllwing best-fitted expressins are fund. b D.7 D k D.39 D D D -.388.484 x w. 5.973.73-3.4-8 x w -.9 x w.755 () () Fr all cmbinatins f t/x, D / and x/w used in the finite element analysis, b and k are calculated frm eqn () and (), and D eq btained frm eqn(9). T further illustrate the applicability f the empirical expressins, the calculated D eq is substituted int eqn (5) t btain chlride cncentratin alng the ack under varius situatins. The results are then cmpared directly t thse frm the finite element analysis. Fig.9 (a) shws the variatin f chlride cncentratin with distance frm the surface at different times, while Fig.9 (b) shws the variatin f chlride cncentratin with time fr different lcatins alng the ack. It is clear frm bth figures that results frm empirical equatins (cntinuus line) are very clse t the finite element results (pen dts). As shwn in Fig.9 (c), fr cases with high ack diffusin cefficient D and large ack width w, the chlride cncentratins frm the empirical mdel are lwer than numerical results at lcatins far away frm bundary with cnstant chlride cncentratin. This is caused by the reflectin effect f the bttm bundary in the numerical simulatins. In the empirical fitting, nly data withut the reflectin effect is selected, s the predictins frm the empirical expressin shuld be free f bundary reflectin effects and mre accurate. (a) (b) (c) C /C C /C C /C.9.8.7.6.5.4.3...9.8.7.6.5.4.3...9.8.7.6.5.4.3..,5,3,,.5 (year) C /C -x D =8 - m /s = - m /s w=6 m FEM Empirical 4 6 8 4 6 8.5,,3,5, (cm) C /C - t D =8 - m /s = - m /s w=6 m FEM Empirical 4 36 48 6 7 84 96 8 T(mn.),3,5,8, (cm) C /C -t D = - m /s =. - m /s w=5 m FEM Empirical 4 36 48 6 7 84 96 8 T(mn.) Fig.9 Cmparisn f chlride cncentratin frm empirical mdel(cntinuus line) with finite element analysis (disete pints) (a) Chlride cncentratin alng the ack at different times; (b) and (c) Chlride cncentratin in the ack at different lcatins vs time 3.4 Time t steel crrsin initiatin As chlride at the steel surface reaches a itical cncentratin, crrsin starts t ccur. The determinatin f crrsin initiatin time 9

is imprtant in the durability assessment f reinfrced cnete structure. Knwing the itical chlride cncentratin C ini and the surface cncentratin C, eqn (5) can again be emplyed t btain the time fr C ini t be reached at a particular distance x frm the surface (s steel with such a cver will start t rust), r the depth f penetratin f the crrsin frnt (which is the distance frm surface where C ini is just reached) at a given time. In the calculatin, D eq in eqn (5) are btained frm eqns (9) t (). Due t the cmplex frm f the empirical equatins, eqn (5) cannt be transfrmed t an explicit analytical frm. It is therefre necessary t slve it implicitly thrugh numerical iteratin. Age (mn.) 8 96 84 7 6 48 36 4 x - t FEM Empirical D =8 - m /s = - m /s w=6 m R =.974 D = - m /s =. - m /s w= m D = - m /s =5 - m /s w= m R =.9999 3 4 5 6 7 8 9 Fig. Time t crrsin initiatin fr varius distances frm the surface 5 9 6 D = - m /s =5 - m /s w= m R =.9997 D =8 - m /s = - m /s w=6 m R =.9986 3 t - x FEM Empirical 4 36 48 6 7 84 96 8 Age (mn.) Fig. Penetratin depth f crrsin frnt at different times Fig. and Fig. shw respectively the calculated results fr crrsin initiatin time at varius distances and depth f penetratin frnt at different times. In these calculatins, C ini /C is taken t be 5%. In bth figures, the cntinuus line represents the result btained frm the empirical mdel and the dts are finite element results. The gd general agreement between the tw can again be bserved. Fr the case with D = - m /s, =. - m /s and w=μm in Fig., the predicted crrsin initiatin time btained frm the empirical mdel is higher than that frm the finite element analysis at lcatins far away frm surface (x>6 cm). This is again due t the reflectin effect at the bttm f the finite element mesh which ineases the chlride cncentratin and hence reduces the time fr itical chlride cncentratin t be reached. Accrding t the results in Fig., the crrsin initiatin time can inease significantly with distance frm the surface. This implies a thicker cver is still effective in delaying steel rusting even when the cnete structure exhibits surface acks. 4 CONCLUSION Fr acked cnete structures expsed t chlride envirnment, chlride diffusivity thrugh the ack (D ) is higher than that thrugh the cnete matrix ( ). As a result, steel crrsin in the acked member will ccur at an earlier time. The analysis f chlride penetratin thrugh the ack is a cmplicated prblem as diffusin alng the ack is cupled with diffusin perpendicular t the ack resulted frm the lateral cncentratin gradient between the ack and the surrunding matrix. T btain a gd understanding f the diffusin prcess, finite element analysis is first carried ut t btain the chlride cncentratin distributin at different times fr cnete with different, D and ack width (w). Then, after a sufficiently large amunt f numerical

data are prduced frm the simulatin, an equivalent diffusin cefficient D eq is derived frm empirical fitting, assuming the diffusin t be ne-dimensinal in nature. D eq btained in this manner is a functin f, w, D as well as time and distance frm the surface. Accrding t dimensinal analysis, D eq can be expressed in terms f three dimensinless parameters. Using the empirically determined D eq, predicted chlride prfile, crrsin initiatin time and penetratin f crrsin frnt are all in gd agreement with finite element results. The apprach develped in the present study therefre prvides a simple but accurate means fr the analysis f chlride diffusin in acked cnete members. REFERENCES [] Andrade, C. Calculatin f Chlride Diffusin-Cefficients in Cnete frm Inic Migratin Measurements. Cement and Cnete Research, 993, 3(3): 74-74. f cntinuus acks n the steady-state regime, Cement and Cnete Research,, 3(): 37 43. [6] Olga, G.R. and Htn, R.D. Influence f Cracks n Chlride Ingress int Cnete, ACI Materials Jurnal, 3, (), -6. [7] Djerbi, A., Bnnet, S., Khelidj, A. and Barghel-buny, V. Influence f Traversing Crack n Chlride Diffusin int Cnete, Cement and Cnete Research, 8, 38(6), 877-883. [8] Chngshi Wu. Methds f Mathematical Physics. Peking University Press,, 8-83.(in chinese) [9] Frank R. Girdan, Maurice D. Weir and Willian P. Fx. A first curse in mathematical mdeling (third editin), China Machine Press, 3, 34-37. [] Luping Tang, Lars-Olf Nilssn. Rapid Determinatin f the Chlride Diffusivity in Cnete by Applying an Electric Field. Materials Jurnal, 993, 89(): 49-53. [3] Arvind K. Suryavanshi, R. Narayan Swamy, Gerge E. Cardew. Estimatin f Diffusin Cefficients fr Chlride In Penetratin int Structural Cnete. Materials Jurnal,, 99(5): 44-449. [4] Aldea, C.-M., Shah, S.P. and Karr, A. Effect f Cracking n Water and Chlride Permeability f Cnete, ASCE Jurnal f Materials in Civil Engineering, 999, (3), 8-87. [5] B. Gérard, J. Marchand, Influence f acking n the diffusin prperties f cement-based materials, Part I: influence