Modelling of Crack Width in Concrete Structures Due to Expansion of Reinforcement Corrosion

Transcription

1 1DBMC International Conférene On Durability of Building Materials and Components Modelling of Crak Width in Conrete Strutures Due to Expansion of Reinforement Corrosion J.J. Zheng 1, C.Q. Li 2 and W. Lawanwisut 2 1 Zhejiang University of Tehnology, The College of Civil Engineering and Arhiteture, Hangzhou, Zhejiang, 3114, P.R. China 2 University of Dundee, Division of Civil Engineering, Dundee, DD1 4HN, UK jjzheng4335@sina.om TT1-89 ABSTRACT Crak width is a parameter of the most pratial signifiane for the design and assessment of reinfored onrete [RC] strutures. Pratial experiene and observations suggest that orrosion affeted reinfored onrete strutures are more prone to raking than other forms of strutural deterioration. Determination of orrosion indued rak width is essential to the predition of the servieability of orrosion affeted RC strutures and to the instigation of repairs for the strutures. Although onsiderable researh has been undertaken on orrosion indued raking proess, a study of researh literature suggests that little work has been arried out diretly on orrosion indued rak width both numerially and experimentally, and no analytial model for the rak width has been published to date. Moreover, for orrosion affeted RC strutures, repairing osts due to onrete raking and spalling exeed those from other forms of deterioration by a substantial margin. It is therefore imperative to aurately predit the rak width in order to ahieve the ost effetiveness in the asset management of RC strutures. This paper attempts to investigate the orrosion indued raking proess in an analytial manner, in whih a theoretial model for orrosion indued rak width in RC strutures is derived. In this paper, frature mehanis is employed for the analysis of stress and strain in the onrete surrounding the reinforing bar. The proposed model is onsidered the volumetri expansion of the ing reinforement through the material moduli. The validity of the derived stress equation is demonstrated by onsidering the model as a thik-wall ylinder subjeted to an internal radial pressure distribution. As the orrosion progresses, the indued stress surrounding the rebar and rak evolution are determined. The predition of the times to onset of raking of the over will be based on a ombination of an expansive pressure analogy and a frature mehanis approah. A merit of the derived model is that it is diretly related to ritial fators that affet the orrosion indued raking proess, suh as the orrosion rate, onrete geometry and property. The derived model is also verified with both experimental and numerial data obtained from researh literature. The model derived in the paper an serve as a useful tool for engineers, operators and asset managers in deision-making regarding the maintenane and repairs of orrosion affeted RC strutures. KEYWORDS Conrete struture, Crak width, Reinforement orrosion, Servieability.

2 1DBMC International Conférene on Durability of Building Materials and Components 1 INTRODUCTION Corrosion of reinforing steel in onrete is the predominant ausal fator in the premature degradation of reinfored onrete [RC] strutures [Broomfield 1997]. Pratial experiene and observations suggest that, although many RC strutures are seen as badly deteriorated, haraterized by mass onrete raking and spalling, they are still struturally sound [Dhir & MCarthy 1999]. The reason for this is attributed to the nature of the problem; the orrosion produts exert an expansive stress on onrete the tensile strength of whih is usually low. It is also partially due to the fat that the safety fators used in strutural design for strength are usually larger than those for servieability sine the paramount importane of strutural safety. As a result, orrosion affeted RC strutures are more prone to raking [than, e.g., loss of strength], inurring onsiderable osts of repairs and inonveniene to the publi due to interruptions. This gives rise to the need for thorough investigation on orrosion indued raking proess in order to ahieve ost-effetiveness in maintaining the servieability of the RC strutures. Considerable researh has been undertaken on orrosion indued raking proess, with perhaps more numerial and experimental investigations than analytial ones. Numerial investigations use mainly finite element methods with various models for the growth of orrosion produts [i.e., expansive pressure] and onrete behavior one raked. For example, Dagher & Kulendran [1992] and Pantazopoulou & Papoulia [21] assume that the raks in onrete are smeared and employ frature mehanis to determine the stress in onrete and hene the raking. Molina et al. [1993] model the onrete as linear softening material and the orrosion produts as a layer with redued modulus of elastiity. With this model, the length of the rak evolution in onrete an be determined. Noghabai [1996] and Coronelli [22] also ombine the orrosion-indued pressure [stress] with the pressure produed by bond ation in determining the stress in onrete. Ueda et al. [1998] use finite element method to examine the fators that affet orrosion indued raking in onrete and find that the tensile strength and reep of onrete are important fators. In experimental investigations on orrosion indued raking in onrete, the orrosion proess is usually aelerated by various means so that onrete raking an be ahieved in a relatively short time [Alonso et al. 1998; Andrade et al. 1993; Liu & Weyers 1998; Franois & Arliguie 1998]. Most of the experiments appear to fous on surfae raking of onrete rather than on diret measurement of rak width over time [Liu & Weyers 1998]. One experimental results are produed, empirial models an be readily developed for orrosion indued onrete raking based on mathematial regression, inluding both deterministi models, suh as Alonso et al. [1998], Rodriguez et al. [1996], and probabilisti models, suh as Thoft-Christensen [21] and Vu & Stewart [22]. The fators that affet the orrosion indued raking have also been studied in experiments. For example, Alonso et al. [1998] find that, in addition to onrete properties, the orrosion rate and over to bar diameter ratio are ritial fators. Determination of orrosion indued rak width is essential to the predition of the servieability of orrosion affeted RC strutures and to the instigation of repairs for the strutures. Although onsiderable researh has been undertaken on orrosion indued raking proess, a study of researh literature [see about referenes] suggests that little work has been arried out diretly on orrosion indued rak width both numerially and experimentally, and no analytial model for the rak width has been published to date. While aknowledging that empirial models exist, e.g., Alonso et al. [1998], for widespread appliation of raking models to orrosion affeted RC strutures by a variety of users, it is more desirable to have analytial models [e.g., Bažant 1979; Liu & Weyers 1998]. Moreover, for orrosion affeted RC strutures, repairing osts due to onrete raking and spalling exeed those from other forms of deterioration by a substantial margin [Dhir & MCarthy 1999]. It is therefore imperative to aurately predit the rak width in order to ahieve the ost effetiveness in the asset management of RC strutures. It is in this regard that the present paper attempts to investigate the orrosion indued raking proess in an analytial manner, in whih a theoretial model for orrosion indued rak width in RC strutures is derived. In this paper, frature mehanis is TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

3 1DBMC International Conférene on Durability of Building Materials and Components employed for the analysis of stress and strain in the onrete surrounding the reinforing bar. Corrosion indued raks in onrete are assumed to be smeared and the onrete is onsidered to be quasi-brittle material. A merit of the derived model is that it is diretly related to ritial fators that affet the orrosion indued raking proess, suh as the orrosion rate, onrete geometry and property. The derived model is also verified with both experimental and numerial data obtained from researh literature. 2 CORROSION INDUCED CONCRETE CRACKING As is well known, onrete with embedded reinforing steel bars an be modeled as a thik-wall ylinder [Bažant 1979; Tepfers 1979]. This is shown shematially in Fig. 1[a], where D is the diameter of reinforement bar, d is thikness of the annular layer of onrete pores [i.e., a pore band] at the interfae between the reinforing bar and onrete, and C is the onrete over. Usually d is onstant one onrete has hardened. The inner and outer radii of the thik-wall ylinder are a = (D+2d )/2 and b = C + (D+2d )/2. When the reinforing steel orrodes in onrete, its produts [i.e., s, mainly ferrous and ferri hydroxides] fill the pore band ompletely. As the orrosion propagates in onrete, a ring of orrosion produts forms, the thikness of whih, d s (t) [Fig. 1[b]], an be determined from [Liu & Weyers 1998] W ( t) d s ( t) = π ( D + 2d 1 ) ρ α ρ st [1] where α is a oeffiient related to the type of orrosion produts, ρ is the density of orrosion produts, ρ st is the density of the steel and W (t) is the mass of orrosion produts. Obviously, W (t) inreases with time and an be determined from [Liu & Weyers 1998] W 1/ 2 t ( t) = 2.15(1/ ) ( ) α πdiorr t dt [2] where i orr (t) is the orrosion urrent density [in µa/m 2 ] whih is a measure of orrosion rate. [a] [b] [] Figure 1. Shemati representation of raking proess. The growth of the ring of orrosion produts [known as band] exerts an outward pressure on the onrete at the interfae between the band and onrete. Under this expansive pressure, the onrete ylinder undergoes three phases in terms of raking: [i] no raking; [ii] partially raked; and [iii] ompletely raked. In the phase of no raking, the onrete ylinder an be onsidered to be elasti isotropi so that the theory of elastiity an be used to determine the radial stress σ r (r) and tangential stress σ (r) at any point ( r ) in the ylinder [Timoshenko & Goodier 197]. From the θ TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

4 1DBMC International Conférene on Durability of Building Materials and Components radial stress σ r (r), the expansive pressure at the interfae between the band and onrete an be obtained as Eef d s ( t) P1 = σ r ( a) = [3] 2 2 b + a a + υ 2 2 b a where E ef is the effetive elasti modulus of onrete and υ is Poisson s ratio of onrete. It may be noted that Equation [3] is the same as that was reported in Liu & Weyers [1998]. From the tangential stress σ (r) at r = a, the initial raking time an be determined by satisfying the ondition σ θ ( a ) = θ f t, where f t is the tensile strength of onrete. After raking initiation, the rak in the onrete ylinder propagates along a radial diretion and stops arbitrarily at r [whih varies between the radii a and b] to reah a state of self-equilibrium. The rak divides the thik-wall ylinder into 2 o-axial ylinders: inner raked and outer unraked ones, as shown in Fig. 1[]. For the outer unraked onrete ylinder, the theory of elastiity still applies. For the inner raked onrete ylinder, let it now be assumed that the raks are smeared and uniformly distributed irumferentially in the raked ylinder [Pantazopoulou & Papoulia 21]. Also let it be assumed that the onrete is a quasi-brittle material. With these assumptions, frature mehanis an be applied to determine the stress distribution in the raked ylinder [Kanninen & Popelar 1985]. Aording to Bažant & Jirasek [22] and Noll [1972], there exists a residual tangential stiffness in the raked onrete. Sine the residual tangential stiffness at eah point on the raked surfae along the radial diretion is dependent on the tangential strain of that point, it is a funtion of the radial o-ordinate r. In view of the lak of knowledge of the residual stiffness of the raked onrete, it will be assumed in this paper that the residual tangential stiffness is onstant along the raked surfae, i.e., on the interval [ a, r ], and represented by α Eef, where α [ < 1] is tangential stiffness redution fator. Based on Bažant & Planas [1998], the stiffness redution fator α is dependent on the average tangential strain ε θ over the raked surfae and an be determined as follows [also see Sheng, et al. 1991] f α = t exp E [ γ ( εθ εθ )] ef ε θ [4] where ε θ denotes the average tangential raking strain and γ is a material onstant. The raking in radial diretion makes the onrete an anisotropi material loally in the viinity of raks. That is, the elasti modulus in the radial diretion is different from that in the tangential diretion. Li et al. [23] developed a formula for rak width based on this onept of frature mehani and the well-known of thik-wall ylinder as shown in Fig. 1. The derivation of rak width formulation is referred to Li et al. [23]. The orrosion indued onrete rak width ( w ) an be expressed as follows w 4πd ( t) 2πbf s t = [5] α α (1 υ E )( a / b) + (1 + υ )( b / a) ef In Equation [5], the key variables are the thikness of orrosion produts d s and the stiffness redution fator α. d s is diretly related to the orrosion rate as shown in Equations [1] and [2]. α is related to onrete geometry and property. Obviously, with the aumulation of orrosion produts, the rak width inreases. This makes sense both theoretially as shown in Equation [5] and pratially as experiened and observed [Andrade et al. 1993; Liu & Weyers 1998]. It needs to be TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

5 1DBMC International Conférene on Durability of Building Materials and Components noted that, due to the random nature of rak ourrene, there may be more than one rak ourring either simultaneously or within a short period of time. In this ase, the assumption that the rak width of all raks is equal ould be made aording to Molina et al. [1993]. Thus Equation [5] is still appliable but w should be divided equally by the number of raks. In any event, Equation [5] represents the maximum rak width on the surfae of onrete. By using the values of basi variables in Table 1 for illustration, the size of a typial rak as a funtion of time an be determined using Equation [5] and shown in Fig. 2. As an be seen the rak width inreases with time as expeted. At the time that the onrete ylinder ompletely raks, i.e., at time to surfae raking, there is an abrupt inrease in rak width, whih reflets the assumed quasi-brittle nature of the onrete..6.5 Crak width (mm) Time (year) Figure 2. Corrosion indued rak width ( w ). Symbol Values Soures C 31 mm Li (23) D 12 mm Li (23) d 12.5 µm Liu & Weyers (1998) E GPa Li (23) f MPa Li (23) t i orr.3686ln(t) µa/m 2 Li (23) α.57 Liu & Weyers (1998) υ.18 Liu & Weyers (1998) ρ 36 kg/m 3 Liu & Weyers (1998) ρ 785 kg/m 3 Liu & Weyers (1998) st Table 1. Values of basi variables used in raking omputation 3 MODEL VERIFICATION As disussed in previous setions, most of the urrent researh on the orrosion indued raking proess fouses on orrosion indued surfae raking [e.g., Liu & Weyers 1998; Pantazopoulou & Papoulia 21]. For this reason, data on time to surfae raking were olleted from the literature. To investigate the time to surfae raking of onrete strutures damaged by orrosion indued internal pressure, Liu & Weyers [1998] arried out a omprehensive experiment on RC slabs subjeted to hloride indued orrosion. They observed the surfae raking behavior of orrosion affeted RC slabs with various onrete geometry and properties [up to five years]. Their results for time to surfae TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

6 1DBMC International Conférene on Durability of Building Materials and Components raking are shown in Fig. 3. Using the same values of their test variables, the alulated time to surfae raking from the proposed model is also shown in Fig. 3. As an be seen, the analytial results are in good agreement with experimental results, with a maximum differene of about 1% for a range of different onrete overs. 4 Time to Surfae Craking (year) Liu & Weyers [1998] Analytial Cover (mm) Figure 3. Experimental verifiation of time to surfae raking. As noted earlier, Pantazopoulou & Papoulia [21] developed a numerial algorithm to determine the time to surfae raking of onrete strutures subjeted to hloride indued orrosion. In their algorithm, the problem of orrosion indued onrete raking is modeled as a boundary value problem and solved using a finite differenes method. Also the raks in the onrete are assumed to be smeared and the onrete is assumed to be quasi-brittle and anisotropi material. These are the same assumptions adopted in this paper and hene their results an be used for omparison. As shown in Table 2, the differene between the numerial and analytial results for rak width is about 1%. Model Time to surfae raking [in year] Differene [in %] Analytial Experimental (Liu & Weyers 1998) Numerial (Pantazopoulou & Papoulia 21) Table 2. Comparison of time to surfae raking A further omparison may be made with experimental results on rak width. It may be appreiated that data on diret measurement of rak widths either from the laboratory or field are sare. Few data are available for laboratory speimens of a pratial size [Vu & Stewart 22]. In this regard, data reported by Andrade et al. [1993] appear to be the only data useable. In their test, the speimens were 15 x 15 x 38 m. The orrosion was aelerated by imposing eletri urrent [as high as 1 µa/m 2 ] so that the measurable rak width an be ahieved within a test period of up to 1 days. Some results of their measured rak width are shown in Fig. 4. Using the same values of their test variables, e.g., orrosion rate, onrete geometry and properties, the alulated rak width is also shown in Fig. 4. As an be seen the analytial results are in reasonable agreement with experimental results. Of TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

7 1DBMC International Conférene on Durability of Building Materials and Components interest here is that almost all measured rak widths are smaller than [or equal to] the alulated rak widths, indiating that the derived model indeed gives maximum rak width as implied in Equation [5]..4 Crak Width (mm) Andrade et al (1993) Analytial Time (year) Figure 4. Experimental verifiation of rak width. 4 CONCLUSIONS An analytial model for orrosion indued rak width in reinfored onrete strutures has been proposed based on the onept of smeared raks and verified against both experimental and numerial results. The model is diretly related to ritial fators that affet the orrosion indued raking proess, namely the orrosion rate, the onrete geometry and property. It an be onluded that the model presented in the paper an predit orrosion indued rak width with reasonable auray and therefore an serve as a useful tool for engineers, operators and asset managers in deision-making regarding the maintenane and repairs of orrosion affeted RC strutures. Timely maintenane and repairs have the potential to prolong their servie life. 5 ACKNOWLEDGEMENTS Finanial support from the Royal Aademy of Engineering, UK with Award No. 1177/93 and Australian Researh Counil with Grant No. LX is gratefully aknowledged. 6 REFERENCES Andrade, C., Molina, F.J. & Alonso, C. 1993, Cover raking as a funtion of rebar orrosion: part I experiment test, Materials and Strutures, 26, Alonso, C., Andrade, C., Rodriguez, J. & Diez J.M. 1998, Fators ontrolling raking of onrete affeted by reinforement orrosion, Materials and Strutures, 31[8], Bažant, Z.P. 1979, Physial model for steel orrosion in onrete sea strutures theory, Journal of Strutural Division, ASCE, 15[ST6], Bažant, Z.P. & Planas, J. 1998, Frature and Size Effet in Conrete and Other Quasibrittle Materials, CRC Press. Bažant, Z.P. & Jirasek, M. 22, Nonloal integral formulations of plastiity and damage: survey and progress, Journal of Engineering Mehanis, ASCE, 128[11], TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,

8 1DBMC International Conférene on Durability of Building Materials and Components Broomfield, J. 1997, Corrosion of Steel in Conrete, Understanding, Investigating & Repair, E & FN Spon, London. Coronelli, D. 22, Corrosion raking and bond strength modeling for orroded bars in reinfored onrete, ACI Strutural Journal, 99[3], Dagher, H.J. & Kulendran, S. 1992, Finite element modeling of orrosion damage in onrete strutures, ACI Strutural Journal, 89[6], Dhir, R.K. & MCarthy, M.J. (Eds). 1999, Conrete Durability and Repair Tehnology, Thomas Telford, London. Fenner, R.T. 1989, Mehanis of Solids, Blakwell Sientifi Publiations, Oxford. Franois, R. & Arliguie, G. 1998, Influene of servie raking on reinforement steel orrosion, Journal of Materials in Civil Engineering, ASCE, 1[1], Kanninen, M.F. & Popelar, C.H. 1985, Advaned Frature Mehanis, Oxford University Press, New York. Lekhnitskii, S.G. 1963, Theory of Elastiity of an Anisotropi Elasti Body, Holden-Day, San Franiso. Li, C.Q. 23, Life yle modelling of orrosion affeted onrete strutures propagation, Journal of Strutural Engineering, ASCE, 129[6], Li, C.Q., Zheng, J.J., Lawanwisut, W. & Chen, Z.J. 23, An analytial model for orrosion indued rak width in reinfored onrete strutures, ACI Strutural Journal (submitted). Liu, Y. & Weyers, R.E. 1998, Modeling the time-to-orrosion raking in hloride ontaminated reinfored onrete strutures, ACI Materials Journal, 95[6], Molina, F.J., Alonso, C. & Andrade, C. 1993, Cover raking as a funtion of rebar orrosion: part 2 numerial model, Materials and Strutures, 26, Noghabai, K. 1996, Environmental effets on bond in reinfored onrete strutures, Pro. of the 7 th Int. Conf. Durability of Building Materials and Components, Stokholm, Sweden, Noll, W A new mathematial theory of simple materials, Arh. Ration. Meh. Anal., 48, 1 5. Plaat, O. 1971, Ordinary Differential Equations, Holden-Day, San Franiso. Pantazopoulou, S.J. & Papoulia, K.D. 21, Modeling over-raking due to reinforement orrosion in r strutures, Journal of Engineering Mehanis, ASCE, 127[4], Rodrigues, J., Ortega, L.M., Casal, J. & Diez, J.M. 1996, Corrosion of reinforement and servie life of onrete strutures, Pro. of the 7 th Int. Conf. Durability of Building Materials and Components, Stokholm, Sweden, Sheng, J. M., Wang, Z.Z. & Jiang, J.J. 1991, Finite Element Methods for Reinfored Conrete and Limit Analysis of Plates and Shells, Tsinghua University Press, Beijing. Tepfers, R. 1979, Craking of onrete over along anhored deformed reinforing bars, Magazine of Conrete Researh, 31[16], Timoshenko, S.P. & Goodier, J.N. 197, Theory of Elastiity, MGraw-Hill Book Company, New York. Thoft-Christensen, P. 21, Corrosion rak based assessment of the life-yle reliability of onrete strutures, Pro. of Int. Cong. Strutural Safety and Reliability, Corotis et al. (eds), Swets & Zeitlinger, 1 7. Ueda, T., Sato, Y., Kakuta, Y. & Kameya, H. 1998, Analytial study on onrete over raking due to reinforement orrosion, Proeedings of an Int. Conf. Conrete Under Severe Condition 2, Tromso, Norway, Vu, K.A.T. & Stewart, M.G. 22, Spatial variability of strutural deterioration and servie life predition of reinfored onrete bridges, Pro. of Int. on Bridge Maintenane, Safety and Management - IABMAS 22, Barelona, CD-ROM. TT1-89, Modelling of rak width in onrete strutures due to expansion of reinforement orrosion,