David De León, Sergio Díaz Camacho, Carlos Alberto González Pérez
Facultad de Ingeniería, Universidad Autónoma del Estado de México. Coatepec, Toluca, Estado de México, México. Phone: 52(722)2140534 Ext. 1002
Abstract
As the traffic demands grow constantly, and some vehicle bridges deteriorate due to corrosion, bridge agencies require non expensive procedures to support decisions on cost-effective maintenance schedules. In this paper a reliability-based formulation is proposed for the prediction of the next inspection time including the corrosion deterioration and the epistemic uncertainty on the corrosion initiation time. For the identification of the bridge integrity state, where little or no follow up has been previously made, the prediction contains a great deal of epistemic uncertainty. The impact of this uncertainty on the bridge failure probability is appraised. The bridge reliability is calculated by comparing the bridge maximum moment with the bridge capacity which depends on the corrosion on the bridge beams. The bridge failure probability is compared with the acceptable failure probability which is used as the safety threshold and it is calculated from the minimization of the annual expected life-cycle cost. Finally, epistemic uncertainty is introduced in the corrosion initiation time and the optimal next inspection time is obtained.
Keywords: bridge reliability, optimal inspection, corrosion deterioration, expected life-cycle cost.
Próximo tiempo de inspección en puentes de concreto presforzado basado en confiabilidad incluyendo el efecto de deterioro por corrosión
Resumen
A medida que las demandas de tráfico crecen constantemente, y que algunos puentes vehiculares se deterioran debido a la corrosión, las agencias de puentes requieren de procedimientos no tan caros para respaldar decisiones sobre programas de mantenimiento costo-efectivos. En este artículo se propone una formulación basada en confiabilidad para la predicción del próximo tiempo de inspección incluyendo el deterioro por corrosión y la incertidumbre epistémica en el tiempo de inicio de la corrosión. Para la identificación del estado de integridad del puente, en donde se ha dado poco o ningún seguimiento, la predicción lleva una gran cantidad de incertidumbre epistémica. Se evalúa el impacto de esta incertidumbre en la probabilidad de falla del puente. La confiabilidad del puente se calcula comparando el momento máximo en el puente con su capacidad, el cual depende de la corrosión en las vigas del mismo. La probabilidad de falla del puente se compara con su valor aceptable, que se usa como umbral de seguridad y que se calcula de la minimización del costo total esperado en el ciclo de vida. Finalmente, la incertidumbre epistémica se introduce en el tiempo de inicio de la corrosión y se obtiene el próximo tiempo de inspección.
Palabras clave: confiabilidad del puente, inspección óptima, deterioro por corrosión, costo esperado en el ciclo de vida.
En forma revisada el 27 de Mayo de 2013
Introduction
Limit state functions for engineering structures are usually formulated in terms of two opposite concepts: cost and safety. Bridges are no exception and its structural safety depends on uncertain conditions of loading and deterioration. When the bridges have a significant operating time and when the traffic loads are being continuously increased, these uncertainties introduce significant concerns on the bridge safety conditions.
Bridge agencies and operators require, therefore, reliable procedures to schedule maintenance actions for a cost-effective national or regional protection program.
In order to consider these uncertainties within a probabilistic framework, the annual structural reliability is taken as the proper safety measure and the target reliability is used as the target or allowable value.
Several studies have been previously performed to formulate bridge optimal inspection schedules by Frangopol and other authors [1 to 3]. However, the epistemic uncertainty on the bridge damage state has not yet been separated. One of the advantages of treating it separately from the aleatory one is that, by doing so, it is possible to objectively propose bounds for the economical convenience to reduce it by improving the collection of information (quality and quantity). Another one is that conservative decisions may be formulated for risk-aversive owners or managers as they may prefer to handle percentile values, say 90 or 95, of the reliability index instead of just its mean value.
Optimal inspection times may be proposed by comparing the bridge actual reliability with its acceptable value in terms of the failure probability.
The acceptable failure probability is calculated for bridges in terms of the cost of failure consequences for a minimum expected life-cycle cost.
Similarly to what is being done to assess the fatigue life of tubular joints in offshore marine platforms [4], a time variation of the bridge reliability index is obtained through the calculation of the bridge reliability under varied scenarios of corrosion attack.
Corrosion is a major damaging factor for reinforced or pre-stressed concrete bridges as studied by several researchers [8 to 12]. One of the model parameters with most of the epistemic uncertainty is the initiation corrosion time. In this paper, epistemic uncertainty is considered for the bridge performance and its impact on the evolution of bridge reliability is estimated.
Acceptable failure probability
According to previous studies [13-14], the expected life-cycle cost, E[CL], includes the structure initial cost, Ci, and the expected cost of the consequences to exceed the limit state within the nominal structure operating life, E[CF]:
E[CL] = Ci + E[CF] (1)
where:
Ci = C1 - C2 ln(Pf) (2)
C1 and C2 are constants which depend on the structural type and Pf is the probability to exceed the limit state. However, the conventional cost-benefit model considers a no bridge replacement within the bridge lifetime, in case of the bridge damage. The conventional formulation is modified to distinguish the temporary economic loss from the permanent economic loss of the case without replacement. The clarification of the replacement consideration is important because the benefits, from the bridge economic operation, are not lost in the event of bridge failure, but just deferred after its replacement. Therefore, the only loss, from the revenues viewpoint, is the one from the deferral and money value which changes with time. Actually, for typical values of Pf about 0.003, the assumption of single replacement is adequate for practical purposes [15].
If the failure consequences are: the cost of bridge substitution and contents CB, the losses related to human lives CH, the user costs CU and the loss of income due to deferral of the revenues R after the bridge replacement:
E[CF] = PVF[CB + CH + CU]Pf + R* PVF'* Pf (3)
]]> where CB, CH and CU are calculated from standard procedures, PVF is the present worth factor required to update future costs to present value:PVF = [1 - e-rT]/r (4)
PVF = [PVF - T* e-rT]* [1 - e-rΔT]/r (5)
where R is the annual revenues, r is the net annual discount rate, T the bridge nominal operating life and ΔT the bridge reconstruction time.
From the minimization of the expected life-cycle cost,
∂E[CL]/∂Pf = 0, (6)
the acceptable annual bridge failure probability is obtained through:
Pf = C2/[(CB + CH + CU)* PVF + R* PVF'] (7)
The curve, for several costs of consequences, is shown in Figure 1.
Cracking moment capacity without epistemic uncertainty
According to Tuutti [12], the structure lifetime (TVU) is given by the corrosion initiation time (T1) plus the propagation time (T2):
Tvu = T1 + T2 (8)
The Tvu value is fixed by the requirements or necessities of each infrastructure facility (a common target service life for vehicular bridges is about 50 years [16]); therefore, this value is known and T1 can be calculated as follows:
T1 = Tvu - T2 (9)
From Torres and Martinez [17]:
where:
XCRIT: corrosion size needed to crack the concrete cover, in mm, calculated from [11]
]]>
where
C: concrete cover, in mm.
f: steel rebars diameter, in mm.
L: length of corroded bar, in mm.
iCORR: average corrosion rate, in mm/year, based on RILEM [7]
iCORR = CTk0i0 (12)
where,
i0: corrosion velocity estimated for 20°C, in μ/year.
k0: coefficient that considers the concrete water/cement ratio (a/c).
From [17], a parabolic function is approximated to the erf to determine the maximum effective coefficient of chloride diffusion (DEFmax), in cm2/s:
where,
T1: corrosion initiation time expressed in years
Ccrit: critical chloride concentration at the depth of prestressed steel as a percent of the cement weight.
Cs: chloride concentration at the surface, as a percent of the cement weight.
And the effective coefficient of chlorine diffusion is:
where,
a/c: water/cement ratio.
Cf: cementing material, in kg/m3
fa: amount of fly ash, in the case of puzolanic cement, as a percent of the cement weight
t: service time of the structure, in years.
Once the value of T1 was estimated from eq. (9), the variation of the annual reliability index (β) was calculated after corrosion beginning, taking into consideration the lost of pre-stressed steel cross section area due to pitting corrosion as described below.
The steel reduction area due to the bridge corrosion has been previously proposed [18]:
where:
Ar(T): net cross section area of a steel bar corroded at T>T1, in cm2 (see Fig. 2)
]]> D0: initial diameter of steel bar, in cmPmax/Pav: maximum versus average penetration ratio.
For the reduction of tension steel due to the prestressing of the strands at time T > T1, an empirical model was used to determine the behavior of tension stress as a function of time as proposed by Carrion et al. [9]:
s(T) = s0(1 - bT)m (22)
where,
s: tension stress in a given time, en kg/cm2
s0: 14 250 kg/cm2 initial tension stress.
T: time in which reducing of _ is measured after the onset of corrosion in years.
b: 0.01538
According to standard procedures [5-7], the cracking moment capacity is calculated and the statistical properties of random variables are shown in Table 1.
Finally, the bridge is modeled as a single component, being the beam under corrosion the main element, the limit state is described by the event where the beam cracking moment capacity MCR is exceeded by the maximum acting moment Mmax, note that the cracking moment MCR is ruled by the pre-stress force, that is directly proportional to the steel surface [5-7]1;
MCR = M1 + M2 (23)
where M1 = Msw + Mslab; Msw is the moment due to the girder selfweight and Mslab is the moment due to the road concrete slab.
where f'c, is the compressive strength of concrete, Sis, Sic, A, and e are geometric properties of the girder cross section and Pf is the total pre-stressed force that depends of the pre-stressed steel transversal section area as pointed out in reference [6].
The MCR is also associated with the presence of fissures and the reduction of the bending capacity; therefore, the bridge annual failure probability is expressed as:
Pf = P(Mmax ≥ MCR) (25)
Mmax:maximum acting moment, calculated for a simple supported beam by the influence lines method for beams [19], using a vehicle T3-S2-R4 as a design live load [19] (see Table 1).
MCR: beam cracking moment, calculated by standard procedures (eq. 23) in accordance with recommendations of ACI [5], ANIPPAC [6] and RILEM [7].
Epistemic uncertainty on corrosion initiation time for bridge damage condition
Corrosion initiation time (T1) is a very difficult to predict variable and its relevance for the bridge capacity is crucial. One way to deal with this modeling uncertainty is introducing a correction factor, to account for epistemic uncertainty, on the available expressions to predict this time. In this paper, and in a simplified way, a random factor is introduced to modify the corrosion initiation time T1. This factor is assumed to be lognormal with median 1 and proposed coefficient of variation CVeT1
A series of 10,000 corrosion initiation times T1ep is generated by using CVeT1 =0.3, and a series of annual failure probabilities and annual reliability indices are calculated for each year after corrosion initiation. Numerical distributions for Mmax and Mc were built for every year after the corrosion initiation time, and its failure probability is calculated. A Monte Carlo simulation process is used for that purpose.
Application to La Joya bridge in Mexico
La Joya bridge is pre-stressed concrete vehicular bridge located in Mexico State, close to Mexico City. Its main span is 30m and the structural type consists on box-shaped prestressed concrete girders (Figure 3) supported by reinforced concrete piers that rest on reinforced concrete piles. The bridge reconstruction time is ΔT = 2 years.
Discussion
After calculating T1ep, the variation of the pre-stressed cross section area and the value of the reliability index were calculated annually for 15 years, each line showed in Figures 4 and 5 represents a realization conducted (for simplicity only 200 experiments are showed in each Figure). The construction of both Figures, involves the effect of the corrosion process; when time after initiation is nearest to T1, the pre-stressed cross section area is nearer to its original value; therefore, the largest the time after T1, the lowest the cross section area and the reliability index.
The bridge failure probability evolves as the corrosion develops in the girder and, as the bridge is under heavy and frequent vehicles loading, the tension stresses trend to accelerate the cracking process in the girder. The evolution of the bridge reliability index, as observed in Figure 4, may be used to establish a time to visually inspect, at least, the bridge condition. The acceptable value of the annual reliability of 2.78 is reached by the bridge reliability at the time range between 5.5 and 10.5 years, after corrosion initiation, from Figure 5, for the coefficient of variation of corrosion initiation time, of 0.3. Being conservative, one may select to inspect the bridge at the time of 5.5 years. However, other option may be to select the time corresponding to the percentiles 5 or 10, which would lead to the inspection time of 6 or 8 years, respectively, depending on the manager conservative degree or the bridge importance in the national road network.
Cost/benefit studies may be used to weight the cost of narrowing this range against the benefit of making more precise predictions of the bridge inspection time.
The bridge importance will move the acceptable reliability index to higher or lower thresholds. Also, the corrosiveness of the location environment, and stresses demand on the bridge, may produce that the girder deteriorates faster and the recommended time to inspect and potentially maintain the bridge may be shorter.
Improvements and upgrading works to the bridge are not considered here.
The bridge age may play an important role as the corrosion may be initiated faster than for other newer bridges. Further studies, with other spans, ages, and importance level, may help to produce guidelines for bridge maintenance, where corrosion is the major damaging factor, in Mexico.
A procedure to calculate the optimal time for next inspection on bridges has been proposed based on yearly bridge corrosion which reduces its bending capacity.
The optimal schedule considers the acceptable bridge reliability which is calculated as a function of the failure losses, which are a measure of the bridge importance.
The procedure accounts for epistemic uncertainty on the corrosion initiation time and this produces a range of potential times to inspect, which allows for conservative decisions by picking the minimum time.
For the example shown here, the next inspection time should be 5.5 years after the corrosion initiation time, in a conservative scheme.
Nota
1 Since the estimation of MC implies a lot of calculations where the target value is related with very specific geometrical properties of girder cross section, area of pre-stressed steel, etc., the authors of this research work encourage the readers to review references 5-7 for a better understanding of the calculation of MC.
References
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