1. Introduction
Many Mexican cities with a high population density, such as Mexico City and Acapulco City, are located in earthquake-prone regions, making them vulnerable to both infrastructure damage and devastation in terms of human lives and economic losses. Communities take years to recover from the economic and social destruction caused by earthquakes. Therefore, earthquake mitigation is of prime importance in the reduction of both the loss of lives and structural damage. In order to ease the recovery period that comes after a seismic event, it is important to estimate the damage condition from a probabilistic point of view, with the objective of calculating an expected structural damage of local infrastructure, such as bridges, which are vital for rescue operations, transport of materials, and emergency equipment.
Given the different environmental loads that RC bridges are subjected to during their lifespan, their elements present structural deterioration, causing both a decrease in their structural capacity and a modification of their structural reliability. Therefore, it is indispensable to develop approaches that allow estimating the reliability levels of RC bridges. Several researchers have proposed different approaches to evaluate structural reliability.
Based on the above, several researchers have proposed different approaches to evaluating structural reliability. For example, [
1,
2] estimate the probability failure of structures by means of the Monte Carlo simulation technique using FORM and SORM methods; [
3] provide a framework to estimate the time-dependent risk in a multihazard environment in bridges; [
4] present a reliability analysis to predict the probability of failure in bridges using the Markov model; [
5] estimate the probability of failure of railway bridges for high-speed trains; [
6] propose a reliability assessment in steel bridges considering deterioration due to fatigue.
The solution to an engineering problem is not such if it has a short lifespan, or if it generates a structure with overdesigned geometric and physical properties. The high number of uncertainties that engineers face during the design process call for the use of concepts and methodologies of structural reliability. For example, [
7] propose a reliability approach to evaluate the structural condition using distribution functions; [
8] present an approach to estimate the structural reliability in a rock tunnel obtaining the failure probability from the first-order reliability method (FORM); [
9] present a life-cycle management for bridges considering the risk attitude of decision-making; the effect of climate change on performance is also considered. References [
10,
11,
12,
13,
14] present approaches with closed-form mathematical expressions.
One of the main concerns of structural engineers is how to design a structure capable of resisting extraordinary actions. Such structural demands appear unexpectedly, and can affect the structures, causing undesirable behavior of their components. Thus, it is important to know when the structure could present an undesired behavior. Researchers have proposed different approaches as the basis for obtaining exceedance demand rates (demand hazard curves): [
15] perform seismic demand analyses in bridges located in California; [
16] analyze the influence of viscous dampers on the probabilistic seismic performance of buildings; [
17] calculate exceedance demand rates in concrete buildings; [
18] propose demand hazard curves in steel buildings considering the effects of seismic isolation; [
19] present a methodology to calculate demand hazard curves in a nuclear power plant; [
20] present an approach to estimate seismic exceedance demand rates in gravity dams; [
21,
22] compute demand hazard curves in buildings structured with buckling-restrained frames; [
23,
24] propose a methodology to perform demand hazard analyses in steel buildings.
The difference between the present study and the works mentioned is that reliability is expressed in terms of both probability of failure and reliability index, using simplified closed-form mathematical expressions considering both aleatory and epistemic uncertainties. Moreover, demand hazard curves are estimated based on two approaches: (a) using closed-form analytical expressions that consider aleatory and epistemic uncertainties, and (b) using numerical integration. Reliability indicators are estimated in a bridge structure located in Acapulco, Guerrero, Mexico.
2. Reliability Approach
In recent years, various attempts have been made to apply probabilistic techniques to determine reliability indexes. The probability of survival of a certain system can be defined as
[
25,
26], where
is the failure annual rate, and
is the time. Thus, the probability of failure is defined as
[
27].
can then be expressed as
On the other hand, the annual failure rate,
, that considers both aleatory and epistemic uncertainties is obtained by the following closed-form expression [
28]:
where
and
r are shape parameters of the seismic hazard curve,
,
, and
are the elements of the median demand, and
.
. and
are the variances of the natural logarithm of the capacity and demand, respectively. Making the hypothesis that
follows a Poisson stochastic process in the probabilistic context given,
is equal to
, and making an equality with Equations (1) and (2), the following expression is obtained:
Making some algebraic steps, the probability of failure,
PF, that considers the uncertainties related with aleatory and epistemic uncertainties is as follows:
where
where
is a correction factor and
=
. Based on the probability of failure, the reliability index
is as follows:
where
is the standard normal distribution function.
3. Demand Hazard Assessment
The exceedance demand rate or demand hazard curve,
, can be obtained as follows [
28]:
where
represents the derivative of the seismic annual rate of exceedance
is the probability that the demand,
, exceeds a preestablished damage level,
, for a given intensity,
y; that is, the structural fragility. In order to propose a practical solution for Equation (7), [
28] present the following hypothesis: (1) the seismic mean annual exceedance rate,
v(
y), can be described in an intensity region of interest by the function
; (2) the median demand can be estimated as
; and (3) the structural demands are distributed lognormally with its standard deviation of the natural logarithm [
29]. Thus, the exceedance demand rate is
where
is the mean annual rate of exceedance of the minimum acceleration presented in the spectral acceleration hazard curve;
is the spectral acceleration corresponding to a damage level,
;
is the variance of the natural logarithm of the demand given a seismic intensity,
If the epistemic uncertainties associated with demand are considered, the following equation is obtained [
28]:
where
represents the variances of the epistemic uncertainties related to structural demand.
5. Conclusions
An approach was proposed to obtain both the reliability index and the probability of failure that considers epistemic uncertainties. Demand hazard curves are also obtained based on numeric and closed-form expressions. The probability of failure is proposed in a closed-form expression format, which has the following advantages: (a) it can be used for different kinds of structures; (b) it can be adapted for different environmental loads; (c) it calculates the probability of failure with and without the consideration of the epistemic uncertainties; and (d) it is familiar to structural engineers.
The approach was illustrated in a continuous bridge designed to comply with a drift threshold equal to 0.004. Uncertainties related to mechanical and geometric properties were considered, together with uncertainties for seismic loadings. Prior to obtaining demand hazard curves, different fragility curves were generated considering different thresholds. Demand hazard curves were determined for drift thresholds between 0.001–0.012. Considering the stipulations in the AASHTO design code [
39], the lifespan of bridges must be guaranteed up to 75 years. Based on the results, the serviceability limit state could be exceeded 17 years before reaching the structure lifespan, which means that the design drift threshold is expected to be exceeded at 58 years. Therefore, the use of a design drift threshold equal to 0.004 is not recommended. A drift threshold exceedance does not mean that the structure is unable to resist seismic loadings; such exceedance means that the structure could present undesirable reliability levels before the recommended interval of serviceability. The reliability index target for bridges is to overcome
= 3.5, considering the structure as new [
39]. Considering such target, the bridge under study presented a reliability index 18.24% lower than the recommendation given by [
38] when epistemic uncertainties are considered; if such uncertainties are not considered, the reliability index is 16.28% lower. Thus, the value of 0.004 of design drift threshold is not recommended for this type of topology. It is recommended to explore lower values of design drift threshold between 0.001 to 0.003.
The reliability index and its probability of failure give certainty about both the safety level that a structure has under design loads, and the capacity of the structure to present serviceability levels during a certain time interval. The exceedance demand rate provides the instant when the structure reaches a certain level of damage. If the damage level compromises the structural integrity, decisions need to be made about inspection or maintenance actions with the aim to extend the lifespan of the system.