Analysis of time-dependent reliability of degenerated reinforced concrete structure

Durability deterioration of structure is a highly random process. The maintenance of degenerated structure involves the calculation of the reliability of time-dependent structure. This study introduced reinforced concrete structure resistance decrease model and related statistical parameters of uncertainty, analyzed resistance decrease rules of corroded bending element of reinforced concrete structure, and finally calculated timedependent reliability of the corroded bending element of reinforced concrete structure, aiming to provide a specific theoretical basis for the application of time-dependent reliability theory.


INTRODUCTION
mong multiple structures, reinforced concrete structure fully integrates the characteristics of steel reinforcement and concrete and cost little; hence it has been extensively promoted in infrastructure construction of China [1].Scholars in civil engineering field keep searching for the application of new materials and innovation of structural system and seeking breakthrough in structural disaster-resistance analysis and design [2].However, durability and life-cycle performance degeneration of structure is an important issue concerned by civil engineering workers.But so far, research progress in this field is not considerable, which may be correlated to the involvement of multiple subjects and large difficulty [3].Randomness of resistance of reinforced concrete structure in the initial stage of construction and attenuation of resistance under the effect of environment is important characteristics [4].With the development of infrastructure construction in China, deep research on the problem is urgent.Material degeneration and structural aging would happen under the effects of human-made environment or natural environment during long-term application, which can weaken carrying capacity and durability of structure.However, durability of reinforced concrete structure has been ignored for a long time [5].An investigation [6] suggested that, more than 70 billion dollars were caused because of corrosion of various structures in America in 1975, 40% of which is caused by corrosion of steel reinforcement.Besides, Britain spends about 200 thousand pounds on ocean environment corrosion every year.Japan even spends 40 billion yen on the maintenance of house structure every year.Reinforced concrete structure is frequently used in China and structural damage caused by durability has been quite serious.Steel reinforcement corrosion has been one of major reasons for the failure of reinforced concrete structure.As civil engineering in China has entered into the stage of aging, scientific detection and evaluation on such kind of structure is in urgent need.Therefore, it is of great significance to make safety evaluation of structure through predicting structural durability.

A
In recent years, scholars in China and abroad have made a large number of experiments on the factors influencing durability of structure and obtained many durability degeneration models of concrete structure [7].Traditional structural reliability calculation model does not vary along with time and the characteristics of existing structure determines that the calculation of structural reliability using time-invariant model cannot consider the degeneration performance of structure.But this study considered the variation characteristics of structural resistance and load along with time and calculated the reliability based on time-dependent concept.In addition, we find that, most scholars focus on the construction of degeneration model and concern little about degeneration parameters.Actually, degeneration parameter is an important component of degeneration model.Analysis of durability degeneration model and time-dependent reliability of structure is of significance only when correct parameters are used.As to another innovation point, this study gave uncertainty statistical parameters of reinforced concrete structure resistance based on the detailed introduction of degenerated reinforced concrete resistance decrease model and made an analysis on the rules of resistance decrease of corroded bending element.

DEVELOPMENT AND STATUS OF STRUCTURAL RELIABILITY ANALYSIS METHOD
he concept of reliability originated from America.In 1939, National Advisory Committee for Aeronautics proposed the concept of airplane accident rate; then the concept was extensively applied in the field of civil engineering.In 1940s, Pugsley and Freudenthal published a paper titled Safety Degree of Structure.They integrated the theory of structural safety degree and the concept of mathematical statistics and proposed a reliability analysis calculation model.Then earliest reliability calculation model was extensively recognized and studied by other scholars [9].In 1969, Cornell, an American expert, proposed the concept of reliability index, integrated it with structural failure rate and put forward the conversion formula with regard to reliability index and structural failure rate [10].Then Cornell and Ang proposed first-order second-moment method for the calculation of structural reliability, which was extensively applied in engineering.Since then, many scholars kept innovating structural reliability analysis and calculation and proposed many structural reliability calculation methods.Currently, researches on time-invariant theory have been relatively mature.However, time-dependent reliability theory involving multiple factors and more complex calculation remains to be further studied.On account of this, many scholars carried out a series of researches on time-dependent reliability of reinforced concrete structure.In 2008, Czarnecki et al. [11] proposed a time-dependent reliability model of reinforced concrete structure after finding the significant changes of reinforced concrete structure resistance along with the extension of time under corrosion environment and then applied the model for predicting the service time of structure.In 2012, Frangopol [2] applied structural health monitoring information obtained after integrating structure related priori knowledge and using Bayes updating probability to make a quantitative analysis on degeneration performance of structure.In 2014, Madsen [13] proposed a new structural timedependent reliability and sensitivity analysis method.Similarly, many Chinese scholars are also dedicated to promoting the development of structural time-dependent reliability analysis.Through analyzing structural resistance decrease process induced by steel reinforcement corrosion, Chen Shoushan et al. [14] designed a method for evaluating and predicting reliability of structure.By investigating the existing resistance information of a structure and using a large number of measured data, Zhou Yan et al. [15] established a resistance time-independent reliability analysis model.

Resistance Decrease Model of Reinforced Concrete structure
The resistance of reinforced concrete structure depends on calculation model, geometrical parameters and material performance of structure.During the whole service process of structure, concrete carbonization and steel reinforcement corrosion can result in decline of strength of steel reinforcement and concrete, narrowing of cross section of steel reinforcement and deterioration of coordination between reinforced concrete and those adverse effects gradually accumulates as time goes on.Therefore, geometrical parameters and material performance of structure would both be degenerated as time goes on, which can weaken the resistance of structure.On account of this, when we consider the degeneration of resistance of reinforced concrete structure, the resistance decrease model can be expressed as: Where ( ) r P t stands for resistance of reinforced concrete structure, [ ] P X stands for expression equation of reinforced concrete structure resistance, ( ) aj f t and ( ) j d t stands for performance and geometrical parameter of jth material and they T are the functions of service time; and ( ) sj k t stands for coordination coefficient of jth steel reinforcement, and it is also the function of service time.

Uncertainty Statistical Parameters of Resistance
Uncertainty of reinforced concrete resistance includes uncertainty of calculation mode, material performance and geometrical parameters.Uncertainty of calculation mode refers to variability induced by basic assumption adopted by resistance analysis and evaluation, uncertainty of material performance refers to variability induced by durability degeneration, such as concrete carbonization, cross section loss induced by steel reinforcement corrosion, corrosion expansion, cracking and separation of concrete cover and mechanical property degeneration of corroded steel reinforcement.Considering the uncertainty of calculation mode, the resistance of reinforcement concrete structure can be expressed using the following formula ( ) ( ) Where ( ) P t stands for random process of resistance and r K stands for random variable of uncertainty of calculation mode.According to relevant methods of mathematical statistics and considering the uncertainty of model parameters, probability statistic characteristic of resistance can be obtained from the following formula.
( ) ( ) ( ) ( ) where K r  and K r  stand for the average value of random variable r K and variable coefficient of calculation mode uncertainty respectively; ( ) P r t  and ( ) P r t  stand for the function of average value and the function of variable coefficient respectively.It can be known from formula (6), ( 7) and ( 8) that ( ) [ ( ), ( ), ( )] ( ) ( ) ( ) ( ) ( ) ( ) where j Z stands for relevant random variable influencing resistance and ( ) stands for result of partial derivative when the average value was used.
It can be seen from the above deduction process that, some statistical characteristics of basic parameters of resistance needs to be determined at first before acquisition of statistical characteristics of final resistance.Depending on the current test means and research conditions, we can obtain some statistical characteristics of parameters easily.Using mathematical method, we can obtain uncertainty statistical rules of final model.

Statistical Parameters of Resistance of Corroded Bending Element
Bending element is the most common structural element in existing reinforced concrete structure.After years of research on corroded element, the calculation of bearing capacity of corroded bending element has been quite mature.Taking bending element of reinforced concrete as an example, resistance decrease rules of corroded bending element is analyzed in this section.
According to relevant knowledge of concrete structure design, bending capacity of normal section of rectangular section of corroded element can be obtained from the following formulas.
where Y stands for bearing capacity of normal section of corroded reinforcement (kN•m), v f and 0 f stand for compressive strength of concrete axis and yielding strength of reinforcement (MPa); 1  stands for coefficient (the value is determined according to relevant regulations in Code for Design of Concrete Structures); l and b stand for the width of section and effective height of section (mm); se B refers to equivalent section area of tensile reinforcement (mm 2 ) which is calculated using formula (11).
where sj K refers to coordination coefficient of ith reinforcement (decrease of cohesive force is considered), sj B refers to section area of jth tensile reinforcement (mm 2 ), and sj  stands for reduction coefficient of yielding strength of jth reinforcement.
( ) ( ) ( ) 1 2 where fv  and fv  stands for average value and variable coefficient of compressive strength of concrete Through analyzing parameters of mode, we can finally obtain relevant uncertainty statistical parameters of resistance of corroded bending element of reinforced concrete.

Basic Principle of Time Comprehensive Analysis method
ime comprehensive analysis method means to take the whole service period of a structure as a reference time period during analysis of structural reliability and obtain resistance and load model considering the changes of structural resistance and loading during the time period.The whole service period is regarded as a reference time period, and statistical characteristics T of random variable relating to resistance and loading are all considered during the time period.Hence the maximum value of probability in the reference time period can be taken as the value of load.The practical conditions of P (t) and S (t) are shown in Fig. 1.From the perspective the concept of time-dependent reliability, P is a time-dependent function.But in time comprehensive analysis method, P is considered as a definite value and also the maximum value in the whole reference time period.It should be pointed out that, definite value herein does not refer to determined value, but random variable.We can obtain the practical value of P from probability density function fp.In time comprehensive analysis method, failure probability can be obtained using the following formula.

 
min max ( ) In practical application of time comprehensive analysis method, probability density function of ( ) S t can be obtained through longterm observation of data.In this way, max S can be obtained probability distribution function.But it is a pity that, long-term observation data are difficult to be obtained.Therefore, the extreme value distribution can be described by some short-term data.time-dependent reliability can be calculated by first order reliability method after analysis of load and resistance rules of structural element.

Analysis of time-dependent reliability of corroded bending element
A flexural simply supported beam component of reinforced concrete in someplace was taken as an example.Beam span was 6000 mm, spacing was 3,900 mm, and section size was 250 mm × 500 mm.The thickness of concrete cover was 25 mm, concrete strength was C20, and grade II rebar with a diameter of 16 mm was used, with a reinforcement ratio of 1.2%.As to the external environment, relative humidity was 71% and temperature was 13 o C; besides, 2 co k was 1.2 and ce k was 2.0.We know that, the reinforcement began to be corroded 13.7 years ago and the concrete cover began to crack due to corrosion expansion 26.5 years ago, after substituting relevant coefficients according the method stated in literature [16].
Other relevant resistance statistical parameters were as follows: K r The relevant parameters were substituted into the resistance decrease model of corroded bending element of reinforced concrete, and then the curve for time-dependent variation of average value and variation coefficient of resistance could be obtained (Fig. 2 and 3).According to Unified Standards for the Design of Reliability of Building Structure (GB50068-2001), the constant load followed normal distribution; the live load followed the I-type distribution of extreme value and the maximum value in the time interval [0, T] also followed I-type distribution of extreme value.The average value and standard error of live load can be obtained using the following formulas.ln 1.2826 Where T Q stands for the maximum random variable of live load in service period [0, T].According to the existing conditions, the average value and standard error of constant load S G  and S G  were 130.1 kN•m and 9.11 kN•m respectively.The average value of the maximum live load could be calculated using formula (18).
1.17 2.63 ln 1.2826 Standard error S Q T  was 1.17 kN•m.
According to the above conditions, we could obtain the changes of reliability index of corroded bending element of reinforced concrete along with service time of structure by applying time comprehensive analysis method.Results are shown in Fig. 4.

CONCLUSIONS
t is of great significance to study the prediction of structural durability.But there is no relatively mature method which can obtain correct prediction results.This study gave out uncertainty statistical parameters of reinforced concrete structure and calculated time-varying reliability of corroded bending element of reinforced concrete using time comprehensive analysis method.Finally, the variation rules of reliability index of corroded bending element along with the changes of service time of structure were obtained.It is no doubt that the method provides a new idea for the prediction of durability and the research results have great value for the prediction and evaluation of durability of existing structures.As durability degeneration of reinforced concrete structure is a quite complex process, some researches concerning degeneration mechanism has not been broken through.Only few issues were considered for durability degeneration of reinforced concrete in this study, and many problems remain to be solved.

Figure 1 .
Figure 1.Changes of resistance and load.
maximum load effect in the whole evaluation reference period [0, T] and min 0 maximum resistance in different stages in the whole evaluation reference period [0, T].

Figure 2 .
Figure 2. Time varying curve of average value of resistance.

Figure 3 .
Figure 3.Time varying curve of variation coefficient of resistance.

Figure 4 .
Figure 4.The changes of reliability index of corroded bending element of reinforced concrete along with service time of structure.
The simply supported beam bears constant load and live load.The average value and standard error of constant load, i.e., G