International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Evaluation of structural reliability for reinforced concrete buildings exposed to corrosion Carrillo-Bueno, Carlos A.; Ruiz, Sonia E.; Tolentino, Dante Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Evaluation of Structural Reliability for Reinforced Concrete Buildings Exposed to Corrosion Carlos A. Carrillo-Bueno Graduate student, Instituto de Ingeniería, Mecánica Aplicada, Universidad Nacional Autónoma de México, Coyoacán 04510, México, DF, México Sonia E. Ruiz Professor, Instituto de Ingeniería, Mecánica Aplicada, Universidad Nacional Autónoma de México, Coyoacán 04510, México, DF, México Dante Tolentino Associate Professor, Sección de Estudios de Posgrado e Investigación, ESIA Zacatenco, Instituto Politécnico Nacional, Gustavo A. Madero 07320, México, DF, México ABSTRACT: A probabilistic criterion proposed by Cornell et al (2002) and extended by Tolentino et al (2012) is used here to study the effect of long-term material degradation due to corrosion of reinforced concrete buildings located in the Pacific Coast of Mexico. The criterion considers, by means of closed form mathematical expressions, the simultaneous variation of the structural capacity and of the structural demand, over time. The structural reliability is represented in terms of two alternative indicators: a) the expected number of failures over a time interval, and b) the confidence factors as functions of time, within a Demand and Capacity Factor Design format. The reliability indicators are extended here in order to take into account the structural deterioration due to corrosion over a time interval. Both, aleatory and epistemic uncertainties are taken into account. The structural reliability is evaluated for a 4-story building subjected to a set of real seismic ground motions recorded in Acapulco Bay, Mexico.1. INTRODUCTION  Reinforced concrete is one of the most common materials in construction; however, it is susceptible to present unacceptable reliability levels when it is exposed, after an interval of time, to the attack of aggressive agents in coastal areas, to icing or snow, or in sites with high levels of contamination. The factors that affect the corrosion of reinforcing steel are several, including temperature, humidity, ocean acidification, airborne pollutants, etc. Depending on the exposure conditions, each of them can influence the time of initiation and/or progression of corrosion.  The cause of failure of reinforced concrete structures subjected to corrosion is due to the loss of structural capacity resulting from the properties deterioration, which affects the adherence to the concrete, section cracks or even spalling (Andrade et al. (1993); Youping and Weyers (1998); Castaneda et al. (1997)). The annual cost of corrosion worldwide is estimated to exceed $US 1.8 trillion, which translates to 3% to 4% of the Gross Domestic Product (GDP) of industrialized countries (Schmitt, 2009). A measure to reduce the social and economic problems that it involves can be solved by providing reliability-based tools for inspection and maintenance planning of the infrastructure and structures affected by corrosion. The planning will depend, in part, on the estimation of the structural reliability considering that both structural capacity and structural demand varies over time due to the deterioration of the structural members caused by corrosion.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 2. CORROSION OF REINFORCED CONCRETE Corrosion induced by chloride penetration occurs mainly in structures exposed to marine environments. The chloride ions are present in seawater; however, as the wind shifts as breeze, these are deposited in structures close to the seafront.  In sections 2.1 and 2.2 several basic definitions related with the corrosion process are reviewed. 2.1. Time when corrosion begins The penetration of chloride ions in concrete is difficult to model. Commonly, the law of diffusion (Fick's law) is used. If it is assumed that the concentration of chlorides in the concrete surface and the diffusion coefficient Dc for concrete are independent, then it is possible to establish the following:                  22 ,,xtxCDttxCc   (1) where  txC ,  represents the concentration of chloride ions, as a percentage of weight of concrete at a distance x  from the concrete surface, after a time t  to be exposed to the source of chloride; and cD  is the diffusion coefficient of chloride. It is assumed that the corrosion process begins when the chloride concentration at the site where the reinforcement is located, reaches a critical value,crC ; then, the time in which corrosion starts iT  can be calculated as (Thoft-Christensen, 2001):  200124 CCCCerfDdTicrci           (2) where 0C is the balanced concentration of chloride in the concrete surface; iC  is the initial concentration of chloride, both expressed as a percentage of the weight of concrete; d is the coating thickness; and erf  is the error function. 2.1.1. Diffusion coefficient The diffusion coefficient cD  is one of the most important variables in equation 2. In order to make a good approximation of Dc, it is necessary to know the water/cement ratio  cw / , temperature (  ) and additives (Jensen et al., 1999). Thoft-Christensen (2001) proposes the following expression for the diffusion coefficient:    941.1/025.31146.11 cwDc                 22 024./48.4/212.38  cwcw  (3) 2.2. Start of concrete cracking due to corrosion The following simple model to determine the reduction of the diameter of the steel reinforcement bars was proposed by Thoft-Christensen (2001):                icorrcorr Tticdtd  0            (4) where 0d is the initial diameter; corrc is the coefficient of corrosion; and corri  is the average annual rate of corrosion.  When the amount of oxide produced is high, it produces pressure on the walls of the concrete around the reinforcing bars. To know the time when the concrete cracking occurs, Thoft-Christensen (2001) proposes the following expression based on studies by Liu and Weyers (1998):                     rustcritcrak kWt 22 (5) where critW  is the critical mass of rust necessary to produce cracking; rustk  is a factor which is proportional to the annual rate of corrosion corri  and the diameter of reinforcement 0d .            anporousruststeelsteelcrit WWW exp   (6) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3        corrrust idEk 015039.7   (7)               0dtW porrustporous   (8)          critporrustan ttdW 20exp    (9) where steel  represents the density of steel; rust  is the density of rust; 57.0 (Liu and Weyers, 1998); porousW  is the rust volume required to fill the pore; anWexp  is the volume of corrosion necessary to fill the space due to the concrete expansion; port  is the thickness in the area equivalent to a porosity of one; and critt  is the thickness of the expansion when the fracture starts. In Eq. 9:               ctcrit VabbaEcfct 2222 (10)                      2/20 portda   (11)                   2/20 portdcb   (12) where c  represents the coating; tf  is the tensile strength of concrete; cE  is the modulus of elasticity; cV  is the Poisson ratio; a  is the diameter of the reinforcement; and b is the radial distance from the center of the reinforcement to the coating. 3. STRUCTURAL RELIABILITY ASSESSMENT In order to evaluate the structural reliability here the concept of average annual rate of structural failure is used, and it is extended to the number of failures in a time interval considering the deterioration (caused by corrosion) of both structural capacity and seismic structural demand for a given intensity. 3.1. Estimating the number of failures in a time interval The average annual failure rate )(cF  represents here the expected number of times per year that the structural capacity  c , associated with a certain limit state, is exceeded due to the effects of the loads corresponding to seismic events of all possible intensities. )(cF  can be calculated as follows (Cornell, 1968; Esteva, 1968):           0 )()( dyyScPdyydcF  (13) where dyyd )(  is the absolute value of the derivative of the seismic hazard curve; and yScP   is the probability that the structural capacity (corresponding to a limit state) c , is smaller than the structural demand, ,S given an intensity, y . The expected number of failures during a time interval is equal to the annual failure rate integrated over a time interval. If the variation of the structural capacity and the seismic demand for a given intensity is considered, then, the expected number of failures evaluated over the time interval  ttt , is expressed as (Tolentino et al., 2012):             ddsdcdyysfcf SC ),()(  (14) where      ,ySCP   is the conditional failure probability when a given intensity y occurs; )( cfC  represents the conditional density function of the structural capacity at the instant of time  ; and ),( ysf S  is the conditional density function of structural demand for a given intensity, y,  for an instant  . Eq. 14 can be solved by means of numerical integration; however, in this study Eqs. 13 and 14           ,,0 0 0ySCPdyydttEtttF        12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 are solved by means of the simplified approach based on the following assumptions (Cornell et al., 2002): a) the environmental hazard curve, )(y , is represented around the intensity of interest by the function rkyy )( ; where k  and r are parameters that fit the seismic hazard curve; b) the structural capacity is assumed lognormal with median Cˆ , and standard deviation of the natural logarithm equal toCln ; c) the function byaD ˆ , represents the median value of the structural demand, Dˆ , around the intensity of interest,  a  and b  are parameters that fit the structural demand curve . It is considered that the structural demand presents a lognormal distribution with standard deviation of the natural logarithm for a given intensity y, equal to yDln. Taking into account assumptions a, b and c, and introducing the contribution of epistemic uncertainties, the expected value of the number of failures,  ttF , , at the end of a time interval  ttt ,  is given by (Torres and Ruiz, 2007):         tttyDrtCF Cbryktt 2ln22,ˆ ,ˆ2exp,        dUCUDC  222ln (15) where                 bDC tatCy1,ˆ,ˆ )()(ˆ (16) where ,ˆ,ˆ DCy is the environmental load intensity associated with the median capacity, )(ˆ tC , at instant  ;  CyD ˆln2 and  Cln2  are the variances of the natural logarithms of structural demand for a given intensity, yD , and the structural capacity, Cˆ , to the limit state of interest, respectively; and 2 UD and 2 UC are the variances of the epistemic uncertainties associated with structural demand and structural capacity, respectively. In order to obtain the expected number of failures that takes into account the variation in time of both structural capacity and structural demand for a given intensity due to corrosion, it is necessary to make the following assumptions (plus the assumptions a, b and c mentioned before):  d) The median of the structural capacity given a state of corrosion,corrCˆ , varies linearly in time t:                     tCcorr   )(ˆ  (17) e) The median of the structural demand, Dˆ , which consider an state of corrosion in the interval [0,t ) , is given by:                  bcorr ytfeD  )()(ˆ   (18) Taking into account assumptions a, b, c, d and e, and integrating Eq. 15 within the time interval  ttt , , then, the expected number of failures considering the variation of the structural capacity and the structural demand for a given intensity is given by:      tCcorrcorrcorr yDrDCF bryktt ,ˆln222ˆ,ˆ 2exp),(   ttcorrtUCtUDtC  ,22ln2  (19) where       brcorr efftetrbbtt ,           ttxZYXftxZYXf  ;;;)(;;;                    brtt 11    (20) where ),( ttcorrF  is the expected value of the number of failures given a state of corrosion at the end of a time interval  ttt , ; and ),( ttcorr   is 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 the correction factor for the expected number of failures at the end of a time interval  (Tolentino et al., 2012; Tolentino and Ruiz, 2015) which considers the variation of the structural capacity and structural demand, for a given state of corrosion. The implicit hypergeometric function  )(;;; txZYXf  in Eq. 20 is solved as follows (Seaborn, 1991):   2)1(!2)1()1(1)(;;; zCCBBAAzCABtzCBAF                  0 !)()()(nnnnnnzCBA (21) where brA 1; brB ; brC  2 (22, 23 and 24)                        eftftz  )()( (25)              efttfttz  ))(()( (26) where A, C and z can take real values, and B integer values. 3.2. Confidence factor  The Demand and Capacity Factor Design (DCFD) format assumes that F  (annual probability of the performance level not being exceeded) is equal to 0 (a performance objective). Here it is considered that the expected number of failures at the end of a time interval  ttt ,  is equal to a prefixed value of 0  multiplied by the time interval t ; then, the following condition is established:                       tttF  0,   (27) The number of failures corresponding to a confidence level, x, for a certain time interval is given by (Torres and Ruiz, 2007):                 LtxFF Ktttt   exp,ˆ,  (28) where     tUTtUCtUDU brbr   22 (29) Substituting Eqs. 19, 28 and 29 into 27:           2ln2ln22,ˆ2exp)(ˆˆtCyDrcorrcorrtCbrtDtCk      tttbrK corrtUTtx   0,exp  (30) Solving and separating terms, the following is obtained:             rbcorrcorrcorrconf tttDCttcorr ,ˆˆ,0 (31) where                 rbcorr ytfeD  00ˆ   (32)          22 ,ln2exp CUTCbr  (33)          22 ,ln2exp DUyTDbr  (34) where  ttcorrconf ,  is the confidence factor within the time interval  ttt , , which takes  into account a state of corrosion. 4. IILUSTRATIVE EXAMPLE In this section the confidence factor due to corrosion is evaluated for a building located Acapulco, Mexico. It is a 4-story 7-bay reinforced concrete regular structure (see Figure 1). The building was designed in accordance with the Mexico City Building Code (RCDF-2004). A ductility factor Q=3 was used.   Figure 1: Elevation and plant of the structure analyzed. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 4.1. Building Corrosion  As mentioned in Section 2, there are several intervals of time related to corrosion: the first is the time of corrosion initiation iT  (Eqs. 2 and 3). Assuming the following values mmd 50 ,4.0/ cw , C 9.27 , %18.00 C (Castañeda et al. 1997), %15.0crC (IMT, No. 182, 2001), and %0iC ; then, iT  results equal to 46 years The second time of interest is when the concrete begins cracking crackt( , see Eq. 5). Assuming that: 33600 mkgsteel  , cmc 5 ,cmd 9.10  , 37850 mkgsteel  , 9.0corri  (Corrosión Atmosférica, 1999); then, crackt  results equal to 9 years. In the present study it was decided to evaluate the reliability of the structure after 46, 55, 75 years, and 100 years of building construction. 4.2. Evaluation of the structural capacity over time The structural capacity is evaluated here by means of Incremental Dynamic Analysis (IDAs). Figures 2 a, b, c and d show the evolution of the structural capacity of the critical story (third story), for each record (S#), corresponding to the following time intervals: 0, 46, 55, 75 and 100 years, respectively. The IDAs were performed using the modified 2D Drain program (Campos and Esteva, 1997).   a) 0-46 years  b) 55 years  c) 75 years  d) 100 years Figure 2: IDAs results for different time intervals.   It is noticed that figures 2 a, b, c and d show a reduction in both structural stiffness and structural strength at the end of the time intervals considered, due to the corrosion effect. The capacity of each structure was obtained considering that the next step in the analysis presents dynamic instability (Vamvatsikos and Cornell, 2002). The standard deviations of the natural logarithm at the end of the time intervals of interest are equal to 0.15, 0.17, 0.19 and 0.22 for the intervals 0-46, 55, 75 and 100 years, respectively. Figure 3 shows the median value of the structural capacity corresponding to the time intervals of interest. 01234560 0.02 0.04 0.06Sa (T1,5%)/gMaximum drift CAPS1S2S3S4S5S6S7S8S9S10S11S12012340 0.02 0.04 0.06Sa (T1,5%)/gMaximum drifCAPS1S2S3S4S5S6S7S8S9S10S11S12012340 0.02 0.04 0.06Sa (T1,5%)/gMaximum drift CAPS1S2S3S4S5S6S7S8S9S10S11S12012340 0.02 0.04 0.06Sa (T1,5%)/gMaximum drift CAPS1S2S3S4S5S6S7S8S9S10S11S1212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7  Figure 3: Median of structural capacity, Cˆ , corresponding to different time intervals. 4.3. Evaluation of the structural demand for a given intensity, over time The evaluation of the structural demand was performed by means of a non-linear dynamic analysis "step by step" in time. The structure was subjected to twelve ground motions (S1 to S12). Figures 4 a, b c and d show the median values of the structural demand corresponding to 0, 55, 75 and 100 years, respectively. The standard deviations of the natural logarithmDln  resulted equal to 0.18, 0.21, 0.21 and 0.24 for intervals 0-46, 55, 75 and 100 years, respectively.  a) 0-46 years  b) 55 years  c) 75 years  d) 100 years Figure 4: Functions corresponding to the median values of the structural demand.  4.4. Correction factor and expected number of failures over time The expected number of failures at the end of the time intervals of interest (Eq. 19) is presented in Figure 5. The epistemic uncertainties associated with the structural demandtUD and with the structural capacity tUCwere assumed equal to 0.2. The parameters k and r were fitted to the seismic hazard curve corresponding to an intensity associated with the near-collapse limit state (story drift limit 03.0 ). 4.5. Confidence factor over time Figure 6 shows the confidence factor ( ,conf  Eq. 31) normalized with respect to the factor that the structure had when it was constructed )( 0, tconf . The figure shows that the behavior of the confidence factor presents the following four major stages: 0.0400.0420.0440.0460.0480.05040 60 80 100Drift of the third storyTime (years)0.000.010.020.030.040.050.0 0.5 1.0 1.5 2.0 2.5Medians of the maximum interstory drift of the third storySa (T1,5%)/g0.000.010.020.030.040.050.0 0.5 1.0 1.5 2.0 2.5Medians of the maximum interstory drift of the third storySa (T1,5%)/g0.000.010.020.030.040.050.0 0.5 1.0 1.5 2.0 2.5Medians of the maximum interstory drift of the third storySa (T1,5%)/g0.000.010.020.030.040.050.0 0.5 1.0 1.5 2.0 2.5Medians of the maximum interstory drift of the third storySa (T1,5%)/gtEC  *4082.40671.0ˆ1tEC  *5760.70488.0ˆ 2tEC  *5105.80491.0ˆ3      33.1/*55*50.601.055ˆ gSaED       33.1/*75*50.601.075ˆ gSaED       33.1/*100*50.601.0100ˆ gSaED     33.1/*01.00ˆ gSaD 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8  Figure 5: Expected number of failures over time  1) From 0 to 46 years the chloride penetration is presented but it does not compromise the structural reliability; 2) in the second stage, about 46 years after building construction, the steel reinforcement has started corrosion; 3) between 46 and 55 years the degradation of the reinforcing steel and coating cracking in reinforced concrete sections is presented; 4) after 55 years, the section is already cracked and the reinforcement continues corroding.    Figure 6: Confidence factor normalized with respect to the initial factor value 5. CONCLUSIONS It was shown the importance of taking into account the phenomenon of corrosion for reinforced concrete structures that are close to a marine environment, where chloride penetration phenomenon takes place. For the example shown, the confidence factor of the building was reduced half of its original value after 75 years of building construction. 6. ACKNOWLEDGMENTS The first author thanks CONACYT for the economic support during his M. Eng. studies. Thanks are given to DGAPA-UNAM for his support under project PAPIIT-IN102114. 7. REFERENCES Cornell, C.A. (1968)  “Engineering seismic hazard analysis”  Bulleting of the Seismological Society of America, 58, 1583-1606. Cornell, C.A., Jalayer, F., Hamburger, R.O., Foutch D.A., (2002)  “The probabilistic basis for the 2000 SAC/FEMA Steel moment frame guidelines”  ASCE Journal of Structural Engineering, 128, 526-533. Esteva, L. (1968)  “Bases para la formulación de decisiones de diseño sísmico”  PhD thesis, UNAM, Mexico. In Spanish Liu, Y., and Weyers, R.E. (1998).  “Modeling the time-to-corrosion cracking in chloride contaminated reinforced concrete structures”  ACI Materials Journal, 95-M65. Thoft-Christensen, P. (2001)  “What happens with reinforced concrete structures when the reinforcement corrodes”  Conference on “Life-Cycle cost analysis and design of civil infrastructure systems”, 35–46. Tolentino, D., Ruiz, S. E., and Torres, M.A. (2012)  “Simplified closed-form expressions for the mean failure rate of structures considering structural deterioration”  Structure and Infrastructure Engineering, 8(5), 483-496. Tolentino, D., and Ruiz, S. E. (2015) “Time-dependent confidence factor for structures with cumulative damage”, Earthquake Spectra, 31, 441-461. Torres, M.A., and Ruiz, S.E. (2007)  “Structural reliability evaluation considering capacity degradation over time”  Engineering Structures, 29, 2183-2192. Vamvatsikos, D., and Cornell, A. (2002)  “Incremental dynamic analysis”  Earthquake Engineering & Structural Dynamics, 31, 491-514.  00.10.20.30.40.50 20 40 60 80 100Time (years)0.00.20.40.60.81.00 50 100Time (years)F0, tconfconf

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