Probabilistic demand assessment of bridges considering the effect of corrosion and seismic loading over time


 This paper proposes an approach to calculate demand hazard curves considering the effect of both corrosion and seismic loadings over time. The corrosion is defined as the reduction of the cross-sectional area in the reinforced bars of concrete, induced by chloride ions. Three corrosion phases are considered: starting time of corrosion, cracking, and evolution time. Seismic loads are characterized as a stochastic Poisson process. Uncertainties related to the randomness of geometric properties, mechanical properties, and seismic loadings are considered. The approach is illustrated in a continuous bridge designed to comply with a drift of 0.002. The structure is located in Acapulco, Guerrero, Mexico. Fragility curves and demand hazard curves are obtained at 0, 45, 57, 75, 100, and 125 years, based on the global drift. The effect of both corrosion and seismic loadings over time increase the annual rate of demand up to 308% between 0 years (without damage) and 125 years after the bridge construction.


22
Corrosion in reinforced concrete structures is an influential phenomenon, particularly in structures located 23 near coastal areas. Such phenomenon is frequently present in structures exposed to weathering, which creates 24 a tough environmental condition when sea breeze acts. The particles of chloride ions that come from the sea hit 25 the surface of the bridge, until the amount of such particles is enough to permeate through the concrete, 26 eventually reaching the reinforcement steel, starting both the corrosion process and the structural deterioration. 27 Corrosion has an economic and social impact: in The United States, it represents losses which are around  Bertolini (2008) shows that the steel bars of reinforced concrete are protected where ( ≥ ℎ | , ) is the probability that a certain damage threshold is exceeded for a given intensity . 85 Considering both the variation in the structural demand and its probability density function given by corrosion 86 at instant , the exceedance rate of the demand at an instant is as follows: 87 is the probability that the structural demand due to corrosion results greater than 90 a pre-established value of ℎ for an intensity at instant , and ( | ) is the probability density function 91 of the seismic demand due to a state of corrosion at instant, ; it is shown that ( , ) is equal to ( ) when 92 = 1. 93

94
The corrosion in reinforced concrete systems appears given by the alkalinity generated by the 95 accumulation of the passive layer in steel. In case of structures near marine environments, chloride ions 96 penetrate the pores of concrete, until they reach the reinforcement steel. Therefore, the passive layer is removed, 97 causing the beginning of corrosion. 98

Corrosion induced by chloride penetration 99
The effect of the corrosion in concrete structures by chloride penetration is not easy to characterize; it 100 is commonly assumed that chloride ions follow the Fick's Law (Fick 1855). If the diffusion coefficient is 101 considered as independent and the concentration of chlorides on the concrete surface is regarded as critical, the 102 following expression can be generated: 103 where ( , ) represents the concentration of chloride ions as a percentage of the concrete weight at a certain 105 distance of the concrete surface at instant ; is the diffusion coefficient. If the critical concentration of 106 chlorides on the structural element is reached, the time that chloride ions take to reach the steel 107 reinforcement can be determined by the following expression: 108 where is the width of the steel rod, is the distance between the centroid of the steel rod and the coating, and 151 is the thickness that is similar to a value of porosity equal to 1. Once the cross-sectional area begins to 152 degrade, a rust layer is generated around the reinforcing steel. When such rust reaches a critical amount, it 153 generates additional stresses in the concrete that cause it to crack. Thoft-Christensen (2002) proposes an 154 expression to determine the amount of rust as follows: 155 = 0 (16) 156

157
Intensities and waiting times are simulated using the seismic hazard curve SHC, which is considered 158 as a known variable associated with the fundamental period T. A critical damping equal to 5% is assumed. The 159 simulation of intensities is based on the cumulate distribution function (CDF) of the SHC, as follows: 160 ) is the expression that fitted the SHC, and 0 is the seismic intensity 163 necessary to produce structural damage in the structure. In this particular case, 0 = 1 m/s 2 , which is 164 associated with an exceedance rate equal to 0 = 0.07737. á represents the maximum value of seismic 165 intensity in the SHC, and are adjustment constants, and represents all possible seismic intensities in the 166

SHC. 167
The arrival of earthquakes can be characterized as a stochastic Poisson process. Therefore, the arrival 168 time between earthquakes can be characterized by an exponential distribution function (Melchers and Beck 169 2017). Making some arrangements, the waiting time between seismic occurrences is: 170 where represents random numbers between 0 and 1 with uniform distribution. 173 3 Cumulative damage assessment 174 Cumulative damage assessment is done based on the assumptions related with the type of structure, 175 loadings, and the existence of a maintenance plan. In this study, the structure is subjected to seismic loadings 176 and corrosion; no maintenance actions are made after loadings. The damage that the structure can gradually 177 accumulate is estimated considering the simulation of seismic intensities and waiting times, as well as the 178 different times related with the corrosion process. The cumulative process of damage is described in Fig. 1. Simulation of intensities and waiting times associated with the n simulated models are generated

Yes
The i-th and (i+1)-th intensities at the instant t0 + Δti and t0 + Δti+1 are associated with the n-th structural model Two seismic records are associated with the i-th and (i+1)-th intensities The records are scaled by a factor ψe=isim/iT which is the result of the ratio of the simulated intensity to the spectral acceleration associated with the dominant frequency of the structure The maximum drift, , , of the structure is obtained A random seismic record is selected, which is then multiplied by a scale factor, βm, that produces the value of, , A seismic record, Ski, is obtained, which produces the displacement caused by the i-th and (i+1)-th simulated seismic intensities The record, ri-1, is scaled by a factor ψe=(i+1)sim/(i+1)T which is the result of the ratio of the simulated intensity to the spectral acceleration associated with the dominant frequency of the structure A seismic signal composed by the accumulated seismic record, Sk(i-1), and the seismic record, ri, is obtained The maximum drift, , , of the structure is obtained A random seismic record is selected, which is multiplied by a scale factor, βm, that produces the value of,

238
The uncertainties associated with mechanical and geometric characteristics are essential parameters to 239 assess the structural fragility with the aim to estimate demand hazard curves. Thus, Table 1 shows the 240 mechanical uncertainties and Table 2 shows the geometric uncertainties considered in this study. 241

249
The nonlinear structural dynamic response is obtained when plastic hinges (areas of concentrated 250 plasticity) appear. The following conditions are taken into consideration in the analysis: lateral stiffness is 251 provided by cap beams and columns; the bridge deck only transmits dead loads; the failure mechanism is 252 reached when plastic hinges appear either at the base of all columns or at the ends where columns and cap 253 beams join. Ruaumoko 3D program (Carr 2003) is used for obtaining the nonlinear response. The moment-254 curvature diagram for reinforced concrete is made using both confined concrete (Mander et al. 1988) and the 255 stress-strain model for rebar provided by Rodríguez and Botero (1995). The moment-rotation relationship is 256 estimated with the modified Takeda hysteresis model (see Fig. 5); and are related to the stiffness in loading 257 and unloading cycles, taking values between 0.5 -0.6; the Ramberg-Osgood factor controls the stiffness loss 258 after rebar yielding, and takes values between 1 ≤ ≤ ∞; 0 is the initial stiffness, and is the stiffness in 259 the unloading cycle. 260  Fig. 9 are described below. Fig. 9a shows that the probability of exceeding 0.002 is almost 326 zero for values of Sa/g smaller than 0.10 for 0 and 45 years. On the other hand, the probability of reaching 0.002 327 is near 1 for values greater than 0.55 Sa/g for all cases. An initial cumulative damage is also noted in the cases 328 of 75, 100 and 125 years for 0.05 Sa/g. Fig. 9b shows that the drift threshold of 0.004 is exceeded for values 329 greater than 0.70 Sa/g, and the probability is close to zero in the cases of 0, 45, and 57 years for intensities not 330 greater than 0.15 Sa/g. Fig. 9c illustrates that the drift of 0.006 is exceeded for values greater than 0.7 Sa/g for 331 the cases of 75, 100, and 125 years. The probability of exceeding 0.006 is close to zero for values smaller than 332 0.25 Sa/g for 0, 45, and 57 years. For the case of 75 years, the probability is near zero for values of Sa/g less 333 than 0.15. Fig. 9d shows that the maximum probability of exceeding the case of 0.012 results equal to 0.64, 334 which means that there is a certain probability that the structure collapses at 0.75 Sa/g in all instants. probability that the drift value of 0.004 (service limit state) is exceeded at 75 years, in accordance with the IMT 370 code, is low for values less than 0.2 Sa/g; such limit state is exceeded for values greater than 0.65 Sa/g. 371 Therefore, reinforced concrete bridges could be designed considering a pre-established design drift threshold, 372 because higher intensities are needed to exceed the service limit state. 373 Demand hazard curves were determined for different instants, considering drift thresholds between 374 0.001 and 0.012. For the design drift threshold of 0.002, demand exceedance rates equal to 0.0151, 0.0179, 375 0.0220, 0.0308, 0.0394, and 0.0489 were obtained for instants 0, 45, 57, 75, 100, and 125 years, respectively. 376 On the other hand, values of demand exceedance rates for the case of 0.004 result equal to 0.00476, 0.00555, 377 0.00676, 0.00935, 0.01191, and 0.01467 for instants 0, 45, 57, 75, 100, and 125 years, respectively. Such values 378 represent differences of 116.51%, 141.94%, 196.49%, 250.19%, and 308.16% between 0 to 125 years. If the 379 structure were designed based on a pre-established demand exceedance rate, the final design of the structure 380 would change, whether the effects of cumulative damage were taken into account or not. 381 The proposed approach allows evaluating the conditions in which a structure is after it has been 382 subjected to both seismic sequences and deterioration by corrosion. The information provided by such 383 evaluation is helpful in the decision-making process during the design of new structures. If the drift that 384 represents the exceedance of a given limit state is known, the demand hazard curves will indicate the reserves 385 of structural strength that the structure has in a certain instant. Moreover, the approach presented also provides 386 useful information on the structural performance over time, which can be taken as the basis for maintenance 387 and inspection plans with the aim of preserving the structure with adequate reliability levels. 388 básica Project CB 2017-2018 A1-S-8700. The second and third authors would like to thank both CONACyT 391 and UAM for their economical support during their Ph.D. studies. 392