A new method for estimating the scale of fluctuation in reliability assessment of reinforced concrete structures considering spatial variability

Abstract

The scale of fluctuation (θ) of the material and geometrical parameters is the basis of studying the spatial variability of reinforced concrete structures. In this article, a new estimation method for the scale of fluctuation based on Bayesian information criterion is proposed. And based on the analysis of experimental data recorded on the three 36-year-old beams of the Jianggong Bridge and 246 corroded steel bars, the scale of fluctuation (θ) of concrete compressive strength (f_c) and steel pitting factor (R) are estimated. The theoretical bending moment of three test beams are calculated considering the influence of the spatial distributions of f_c, R, and other relative variables. The reasonableness and superiority of the Bayesian information criterion model than the auto-correlation function method and the semivariogram function model are verified by comparing the theoretical results with the measured bending moment of the three beams mentioned above.

Keywords

Bayesian information criterion bending moment concrete compressive strength reinforced concrete scale of fluctuation steel pitting factor

Introduction

In the past, the study on the reliability of reinforced concrete structures is typically based on the assumption that the material properties are uniform in space, only considering the time-varying and uncertainty of the material or environmental parameters (Dimitri and Leonid, 2009; Frangopol et al., 1997; Guo et al., 2011), and without considering the influence of the spatial variability of the structural performance parameters on the structural reliability evaluation. In fact, more and more indoor experiments and field inspections have proven that there are differences in material properties and structural dimensions at different positions, such as the concrete compressive strength (f_c), the surface chloride content (C_s), and the apparent diffusion coefficient (D_app) (Engelund and Sorensen, 1998; Karimi, 2001; Mullard and Stewart, 2009; Stewart and Suo, 2009), and the reliability of the structure will be overestimated if the spatial variability of the structure is not considered. Therefore, the spatial variability of structural performance parameters is very important for evaluation of the structural reliability.

A parameter considering spatial variability is mainly determined by the mean (μ), the variance (σ), and the scale of fluctuation (θ) (Vanmarcke, 1977). However, there is a limited research on the scale of fluctuation of reinforced concrete structure parameters. Karimi (2001) estimated the scale of fluctuation using the curve-fitting method based on the analysis of C_s data collected from three crosshead beams exposed to de-icing source of chlorides. Li (2004) used the maximum likelihood function to estimate the scale of fluctuation of the surface chloride content and the apparent diffusion coefficient, but this method does not lead to the unique value of θ, and the author mistakenly gave the name “the scale of fluctuation” to the correlation length d (Kenshel, 2009; Li et al., 2004). O’Connor and Kenshel (2013) carried on the simple Kriging method to interpolate the experimental data recorded on a 27-year-old reinforced concrete (RC) bridge located in a marine environment, then estimated the scale of fluctuation of C_s and D_app by the maximum likelihood method and the autocorrelation curve-fitting method, and the results are compared with the corresponding values in Karimi (2001) and Engelund (1997). Stewart (2004) assumed that the values of the scale of fluctuation θ are 3.5 m for all the random field (RF) variables of pitting corrosion based on the limited work about the spatial variability of parameters published by the earlier researchers. Vu and Stewart (2005) studied the corrosion-induced cracking based on the correlation length d = 2.0 m derived from “engineering judgment” for the concrete compressive strength, concrete cover, and surface chloride concentration; however, the calculation process was not described in detail.

At present, only a few scholars estimate the scale of fluctuation based on the measured data of service bridges and mainly focus on the surface chloride content and the apparent diffusion coefficient. There are few studies on the spatial fluctuation of the concrete compressive strength. Meanwhile, most of the researchers estimate the scale of fluctuation of structural dimensions and material properties parameters based on the maximum likelihood function and the curve-fitting method. However, the former estimation method cannot obtain the unique value of θ (Kenshel, 2009). For the latter estimation method, the influence of the correlation function type on the calculation result cannot be ignored. Moreover, due to the lack of large sample data, there is a big difference between the measured data and its fitting curve, and it is hard to determine the appropriate fitting parameters. In other words, the correlation coefficients ρ_exp(τ) calculated by the auto-correlation function (ACF) method contain a large number of negative values, especially in the middle and posterior segment of the fitting curve. While the theoretical value of the commonly used Gaussian model is greater than zero, which makes the measured value cannot be fully utilized in the fitting process, and the reliability of the calculation result is greatly reduced. Therefore, it is necessary to find a more accurate estimation method for the scale of fluctuation.

In order to more reasonably estimate the scale of fluctuation of material and geometrical parameters, in this article, a new estimation method based on Bayesian information criterion (BIC) is proposed. Then, based on the three 36-year-old beams of the Jianggong Bridge, the compressive strength of concrete is measured, and the scale of fluctuation of f_c is estimated. Besides, the spatial properties of pitting factor of 246 corroded steel bars, which are obtained by the accelerated corrosion tests, are estimated. The theoretical values of the bending moment of beams are calculated based on the different θ values of f_c and steel pitting factor R, which are estimated by the BIC method, the ACF method, and the semivariogram function (SVF) method, respectively. Finally, the BIC method proposed in this article is verified by comparing the theoretical results with the experimental results of the bending moment and the failure position of the tested beams.

BIC method

Estimating the value of scale of fluctuation (θ) can be regarded as a model selection problem. According to the previous research results (Akaike, 1980; Honjo and Kashiwagi, 1999), BIC is suitable for Bayesian statistical model identification and is a very useful method for selecting the best model from several alternative models. Therefore, in this article, the scale of fluctuation (θ) is regarded as an unknown parameter and its value can be estimated by maximizing the posterior probability model, while minimizing the value of BIC. The general definition of BIC is as follows

BIC = - 2 \cdot \ln (\overset{\land}{L}) + \ln (n) \cdot k

(1)

where $\overset{\land}{L}$ is the maximized value of the likelihood function (L) of the model for choosing the most appropriate values for unknown parameters, k is the number of unknown parameters in the BIC model, and n is the number of measured data for model calculations.

In order to obtain the expression between the likelihood function (L) in BIC and the unknown parameter (θ), first, the Bayesian estimation of structural parameters considering spatial variability is proposed. It is assumed that the variable Z is a collection of all m variables to be estimated and is defined in a stochastic Gaussian field. Then, the best estimator ( Z *) for a discrete spatial point field at different positions x₁, x₂, …, x_n is

Z^{*} = {[Z_{1}^{*} (x), Z_{2}^{*} (x), \dots, Z_{m}^{*} (x)]}^{T}

(2)

Z_{i}^{*} (x) = {[Z_{i}^{*} (x_{1}), Z_{i}^{*} (x_{2}), \dots, Z_{i}^{*} (x_{n})]}^{T}

(3)

In time t (year), the observation vector Y , which includes the observation data at different positions x₁, x₂, …, x_n can be represented as

Y = {[Y (x_{1}), Y (x_{2}), \dots, Y (x_{n})]}^{T}

(4)

Herein, it is postulated that the observation Y is expressed as a linear function of Z with an observation error α (Rungbanaphan et al., 2010)

Y = {MZ}^{*} + α

(5)

where M is the n×mn coefficient matrix, and α is the Gaussian observation error vector, which is assumed to follow N(0, η ). When the observation errors are independent of each other, $η = σ_{α} I_{n, n}$ , in which σ_α is the variance of the observation errors and I_n,n is the identity matrix.

Based on the prior information of the parameters in the collection Z , a sample field may be simulated by (Hoshiya and Yoshida, 1996)

Z^{*} = Z_{0} + β

(6)

where Z ₀ is the mean of prior information. β is the uncertainty of the prior mean, which is simulated to follow N(0, ω ), in which ω is a covariance matrix for the prior information with the an expression as

ω = σ_{z}^{2} ρ = \begin{matrix} σ_{z}^{2} [\begin{matrix} ρ_{11} & ρ_{12} & \cdot \cdot \cdot & ρ_{1 n} \\ ρ_{21} & ρ_{22} & \cdot \cdot \cdot & ρ_{2 n} \\ \cdot \cdot \cdot & \cdot \cdot \cdot & ρ_{i j} & \cdot \cdot \cdot \\ ρ_{n 1} & ρ_{n 2} & \cdot \cdot \cdot & ρ_{n n} \end{matrix}] & {\begin{matrix} 0 \leq ρ_{i, j} < 1 \\ \begin{matrix} ρ_{i, j} = 1 & i = j \end{matrix} \\ ρ_{i, j} = ρ_{j, i} \end{matrix} \end{matrix}

(7)

where $σ_{z}^{2}$ and ρ are the variance of prior information and the autocorrelation coefficient matrix, respectively. ρ_ij is the value of auto-correlation coefficient between the point x_i and point x_j, which can be calculated using the Gauss autocorrelation function, and then

ρ_{ij} = ρ (τ = | x_{i} - x_{j} |) = \exp [- {(\frac{| x_{i} - x_{j} |}{θ / \sqrt{π}})}^{2}]

(8)

In order to simplify the calculation of the Bayesian estimation, first, σ_α and θ are assumed to be known (however, the actual value is not known), and the Bayesian posteriori model and the Bayesian estimator of the variable Z are obtained. Then, the maximized value of the likelihood function of the BIC model for choosing the most appropriate values for unknown parameters is derived. Finally, the optimal selection of pair (σ_α, θ) is back-estimated based on the BIC method. Based on the Bayesian estimation theory, the posterior probability model of the variable Z is

\begin{array}{l} p (Z | Y, σ_{α}, θ) = \frac{p (Y | Z, σ_{α}) p (Z | θ)}{\int p (Y | Z, σ_{α}) p (Z | θ) d Z} \\ = C_{1} \cdot p (Y | Z, σ_{α}) \cdot p (Z | θ) \end{array}

(9)

{\begin{matrix} p (Y | Z, σ_{α}) = \frac{1}{{(2 π)}^{n / 2} {| η |}^{1 / 2}} \exp [- \frac{1}{2} {(Y - MZ)}^{T} η^{- 1} (Y - MZ)] \\ p (Z | θ) = \frac{1}{{(2 π)}^{n / 2} {| ω |}^{1 / 2}} \exp [- \frac{1}{2} {(Z - Z_{0})}^{T} ω^{- 1} (Z - Z_{0})] \end{matrix}

(10)

where $C_{1} = [\int p (Y | Z, σ_{α}) p (Z | θ) dZ]^{- 1}$ , which is a constant term of a value larger than 0.

Since both the value of σ_α and θ are assumed to be known, the Bayesian estimator of the variable Z , denoted as Z ^* , is obtained by maximizing equation (9), which is equivalent to minimizing objective function Q as follows

\begin{matrix} \max [p (Z | Y, σ_{α}, θ)] = \min [Q (Z | σ_{α}, θ)] \\ = \min [{(Y - MZ)}^{T} η^{- 1} (Y - MZ) + {(Z - Z_{0})}^{T} ω^{- 1} (Z - Z_{0})] \end{matrix}

(11)

But in fact, σ_α and θ are the unknown parameters. So the value of k in the BIC is 2, the likelihood function $(L (σ_{α}, θ | Y))$ and its maximum value $(\overset{\land}{L} (σ_{α}, θ | Y))$ of the model for choosing the most appropriate values of σ_α and θ in BIC are obtained by integrating equation (9). Then, equation (1) can be simplified as

\begin{matrix} BIC & = (- 2) \ln (C_{1} \cdot {(2 π)}^{n / 2} \cdot {| M^{T} η^{- 1} M + ω^{- 1} |}^{- \frac{1}{2}} \cdot p (Z | Y, σ_{α}, θ) |_{Z = Z^{*}} + C_{2}) + \ln (n) \cdot 2 \\ = \ln | M^{T} η^{- 1} M + ω^{- 1} | - 2 \ln p (Z | Y, σ_{α}, θ) |_{Z = Z^{*}} + C_{3} \end{matrix}

(12)

where C₂ and C₃ are the constant terms when the number of samples is determined.

The MATLAB software is used to calculate the above steps, and the optimized selection of pair (σ_α, θ), which is maximizing the value of posterior probability model, while minimizing the value of BIC, is obtained by constantly trying the value of pair (σ_α, θ).

Estimating the scale of fluctuation θ for f_c

Structural description

In order to estimate the scale of fluctuation of structural dimensions and material properties parameters in actual structures, three 36-year-old beams of the Jianggong Bridge in de-icing environment, which is a simply supported bridge located in Ningxiang County, Changsha City, China, were moved to the laboratory for a visual inspection and load-carrying capacity test at the structural testing center in Changsha University of Science and Technology. The length, the width, and the height of the beam are 8.0, 1.05, and 0.66 m, respectively, and the rib width of the beam is 0.15 m. The concrete grade is C30, and the main reinforcements including 4Ф24 and 2Ф22 (middle row) reinforcements are arranged on the lower part of each beam web. The type of stirrup is Ф8, and the spacing is 150 mm. The component information is shown in Figure 1.

Figure 1.

Actual beam specimen of Jianggong Bridge: (a) main beam component and (b) component cross section (cm).

Data acquisition

The concrete compressive strength has a great impact on the load-carrying capacity of actual RC components (Kenshel, 2009), and thus the measured data of the concrete compressive strength is taken as the research object. The core sampling method is used to test the concrete compressive strength (see Figure 2). After the completion of the static load test of the component, 10 core zones on both sides of the beam web are selected for core sampling using HITIDD200 core drilling machine. Then, the samples were processed into small cylinders with a diameter of 100 mm and a height of 150 mm, and a total of 60 specimens were obtained. Finally, the specimens were numbered, and their compressive strength was measured based on the provisions of CECSO03:88. It should be noted that the actual measured axial compressive strength of non-standard size specimen should be corrected by the size conversion coefficient, and the correction factor can be obtained from the above specification.

Figure 2.

Concrete compressive strength testing by the core sampling method: (a) drilling core samples, (b) sample processing, and (c) compressive strength testing.

Based on the standard size of web of the three beams (each beam is divided into A web and B web, as shown in Figures 1(a) and 3. Then, there is a total of six beam webs), the left vertices at the bottom of the lower edge of the beam web is set as the coordinate origin, and a two-dimensional rectangular coordinate system is established, in which the longitudinal direction of beam is X axis and the vertical direction of beam is Y axis. The sample location of partial webs (e.g. sample location of P3-B is shown in Figure 3) and the corresponding inspected results are listed in Table 1. In Table 1, for the data (q,r,s), s represents the compressive strength of concrete (MPa), q is the coordinate X (m), and r is the coordinate Y (m).

Figure 3.

Sample locations.

Table 1.

Partial sample locations and the corresponding concrete compressive strength of beam webs.

Beam web number	Location of measured points and the corresponding compressive strength of concrete
P1-B	Core 1	Core 2	Core 3	Core 4	Core 5
	(0.51,0.332,27.6)	(1.16,0.304,25.4)	(1.74,0.372,30)	(2.92,0.336,24.1)	(3.66,0.364,25.6)
	Core 6	Core 7	Core 8	Core 9	Core 10
	(4.28,0.322,22.8)	(4.98,0.342,27.6)	(5.86,0.361,25.4)	(6.78,0.366,20.6)	(7.48,0.348,27.4)
P3-A	Core 1	Core 2	Core 3	Core 4	Core 5
	(0.82,0.322,24.9)	(1.18,0.342,21.2)	(1.74,0.362,23.2)	(2.36,0.334,19.7)	(3.12,0.398,26)
	Core 6	Core 7	Core 8	Core 9	Core 10
	(3.74,0.346,18.9)	(4.66,0.357,27.9)	(5.25,0.334,18.3)	(6.46,0.318,22.3)	(7.22,0.388,24.1)

Estimated results of BIC method

The priori information in the BIC model can be obtained based on the empirical relationships or the data collected from past projects, and with the increase in the relevant data collected from past projects or the measured data, the result of the BIC method is more accurate, which is one of the advantages of the BIC method. At present, the time-varying model of the relative concrete compressive strength (ratio of the measured value, f_c,_t(x_i), to the initial value, f_c₀(x_i)) of service structures is mainly the quadratic function model, in which the model parameters are assumed to be a, b, and c. Based on the measured data of the service concrete samples with the time span from 1950 to 2010, the relationship between the mean and standard deviation of the relative concrete compressive strength and service age, t, are obtained by Gao et al. (2015). In this article, the research results of the above literature are introduced, and the parameters of BIC model can be expressed as

Z^{*} = {[\underset{n}{\underset{︸}{a^{*} (x_{1}), a^{*} (x_{2}), \cdot \cdot \cdot, a^{*} (x_{n})}}, \underset{n}{\underset{︸}{b^{*} (x_{1}), b^{*} (x_{2}), \cdot \cdot \cdot, b^{*} (x_{n})}}, \underset{n}{\underset{︸}{c^{*} (x_{1}), c^{*} (x_{2}), \cdot \cdot \cdot, c^{*} (x_{n})}}]}^{T}

(13)

Y = {[\frac{f_{c, t} (x_{1})}{f_{c 0} (x_{1})}, \frac{f_{c, t} (x_{2})}{f_{c 0} (x_{2})}, \dots, \frac{f_{c, t} (x_{n})}{f_{c 0} (x_{n})}]}^{T}

(14)

M = [t^{2} \cdot I_{n, n}, t \cdot I_{n, n}, I_{n, n}]

(15)

\begin{matrix} Z_{0} & = {[\underset{n}{\underset{︸}{a_{0}^{*} (x_{1}), a_{0}^{*} (x_{2}), \dots, a_{0}^{*} (x_{n})}}, \underset{n}{\underset{︸}{b_{0}^{*} (x_{1}), b_{0}^{*} (x_{2}), \dots, b_{0}^{*} (x_{n})}}, \underset{n}{\underset{︸}{c_{0}^{*} (x_{1}), c_{0}^{*} (x_{2}), \dots, c_{0}^{*} (x_{n})}}]}^{T} \\ = {[\underset{n}{\underset{︸}{- 3.2 \times 10^{- 4}, \dots, - 3.2 \times 10^{- 4}}}, \underset{n}{\underset{︸}{1.704 \times 10^{- 2}, \dots, 1.704 \times 10^{- 2}}}, \underset{n}{\underset{︸}{0.8401, \dots, 0.8401}}]}^{T} \end{matrix}

(16)

ω = [\begin{matrix} 6.179 \times 10^{- 5} ρ & 0_{n, n} & 0_{n, n} \\ 0_{n, n} & 3.568 \times 10^{- 3} ρ & 0_{n, n} \\ 0_{n, n} & 0_{n, n} & 0.1176 ρ \end{matrix}]

(17)

where f_c₀ is the initial compressive strength of concrete, due to the lack of the initial measured value of the concrete compressive strength at different positions, it is assumed in this study that f_c₀(x_i) is a random number generated by f_c₀ at different positions, 0_n,n is an n×n zero matrix, and the values of the prior information parameters in Z ₀ and ω are both derived from Gao et al. (2015).

Therefore, through the MATLAB software programming, the value of pair (σ_α, θ), which results in the minimum value of BIC model, is obtained, and the estimated values of the scale of fluctuation θ of f_c are shown in Table 2.

Table 2.

Values of θ of f_c.

Beam number	θ for f_c (m)
	BIC	ACF	SVF
P1	2.432	1.456	2.231
P2	2.894	1.048	2.136
P3	3.453	1.690	3.033
Average	2.926	1.398	2.467

BIC: Bayesian information criterion; ACF: auto-correlation function; SVF: semivariogram function.

The above results are the average values of the scale of fluctuation θ of A web and B web on both sides of the beam.

Model verification

As discussed earlier, three 36-year-old beams of the Jianggong Bridge are selected as the research object. Based on the one-dimensional random field (RF) theory, each beam is discretized into several small elements, and the transformation matrix, which is necessary to convert the randomly generated data into a spatially correlated data of f_c, can be obtained through the estimation results of the θ values of f_c. Hence, the fluctuation intensity of f_c along the beam can be described. Then, according to the value of parameters of each discrete element, the bending moment of each element, hence the bending moment of the three beams, can be obtained. Finally, the reasonableness and superiority of the BIC model is verified by comparing the theoretical results of the bending moment with the load-carrying capacity test results of the three beams.

Load-carrying capacity test results

The load tests were conducted to evaluate the bending moment of the three beam specimens, the beams were simply supported over a span of 7.6 m (L), and the four-point loading method is adopted. There were 17 loading steps in the experimental test process. The load-carrying capacity of beam P1, beam P2, and beam P3 are 264, 284.1, and 291 KN, respectively, and the corresponding bending moment at the mid-span section are shown in Table 3. After the completion of the load-carrying capacity test, the concrete compressive strength was obtained by the axial compressive strength test of drilling core samples, and the concrete carbonation depth was tested using the phenolphthalein solvent. Then, the thickness of the concrete cover, the corrosion rate of steel bars, and the mechanical properties of steel bars were also measured. The relevant measured data are shown in Table 4.

Table 3.

Comparison of the theoretical and experimental values of bending moment and the corresponding relative errors.

Bending moment (kN m)		Beam P1	Beam P2	Beam P3
Experimental results in the mid-span section		668.8	719.7	737.2
Theoretical values in the critical section	BIC	642.9	688.3	714.7
	SVF	625.2	669.2	692.8
	ACF	608.3	644.5	665.1
Relative error	BIC	3.87%	4.36%	3.05%
	SVF	6.52%	7.01%	6.02%
	ACF	9.05%	10.45%	9.78%

BIC: Bayesian information criterion; ACF: auto-correlation function; SVF: semivariogram function.

Table 4.

The measured data of material properties.

Beam number	Concrete			Corroded reinforcement
	Ultimate compressive strength (MPa)	Thickness of concrete cover (cm)	Carbonation depth (mm)	Diameter D = 22 mm		Diameter D = 24 mm
				Yield strength (MPa)	Ultimate strength (MPa)	Yield strength (MPa)	Ultimate strength (MPa)
P1	25.6	3.5	23.1
P2	20.4	3.4	22.7	293.4	440.3	252.1	393.3
P3	22.2	3.5	23.3

Theoretical analysis

In order to verify the reasonableness and superiority of the BIC model for estimating the scale of fluctuation of structural dimensions and material properties, the ACF method and the SVF method are selected to estimate the scale of fluctuation of f_c for a comparison. Considering the results of previous related studies, the Gaussian autocorrelation function model and the Gaussian semivariogram model, which have been most frequently used by researchers in the random field analysis of reinforced concrete structures (Gomes and Awruch, 2002; Li et al., 2004; Kenshel and O’Connor, 2009; Vu and Stewart, 2005), are adopted for the ACF method and the SVF method, respectively. Moreover, the sample correlation coefficients ρ_exp(τ) and the sample semivariogram γ_exp of f_c corresponding to different lag distance τ values are calculated according to Ramachandran et al. (2001) and Clark and Harper (2000), respectively. Then, the scale of fluctuation θ of f_c of all beam webs can be estimated by the curve fitting of the analytical model. All the estimated results of ACF method and SVF method are shown in Table 2.

In the theoretical analysis of bending moment of the corroded beams, if there is a positive correlation between the two spatial variables, for example, the concrete compressive strength and the chloride diffusion coefficient, it is reasonable to be assumed that these two variables have the similar fluctuation properties (Kenshel, 2009). Therefore, the θ values of D_app and corrosion rate density (i_corr₍₁₎) can be considered the same as the θ value of f_c obtained in this article. For other spatial variables except C_s (the θ values of C_s is 2.7 (Kenshel, 2009)) and the steel pitting factor R (this will be discussed in later chapter), the θ values can be considered as 3.5 (Vu and Stewart, 2005).

In the estimation of the residual cross-sectional area of corroded reinforcement, the hemispherical pit configuration proposed by Val and Melchers (1997) is adopted. Since the residual cross-sectional area in the case of pitting corrosion is highly dependent on the variation of the pitting factor, R (ratio of the maximum penetration depth and the average penetration depth), the variability and spatial distribution of pits should be further analyzed. Therefore, based on the accelerated corrosion tests on the reinforcing bars with a diameter of 22 mm (Ф22) embedded in the RC beams, the depth of corrosion pits on the corroded steel bars (a total of 246 corroded steel bars with the cross-sectional area loss rate of 3.90% to 20.65%) are measured. The associated measured results of the pit depth are indicated in Table 5. The values of the pitting factor of all reinforcements are statistically analyzed, and the mean and variance of R are 6.574 and 1.994, respectively. Besides, based on the measured data, the spatial distribution of R of each corroded steel bars are studied by the three estimation methods. The results show that the mean and variance of the θ value of R obtained by the BIC method, ACF method, and SVF method are 0.245 and 0.186, 0.197 and 0.131, 0.211 and 0.175, respectively. It is found from the above analysis that the mean value of θ is very small compared with the length of corroded steel bar, and the correlation length d is also very small. Thus, it can be inferred that there is no strong correlation between the spatial distributions of each pit on the surface of a corroded steel reinforcement.

Table 5.

Measured positions and the corresponding pit depth of partially corroded steel bars.

Number of steel bars	(Measured position/cm, the corresponding pit depth/mm)						Cross-sectional area loss of steel bars
L5-B	(18, 1.44)	(35, 1.25)	(49, 1.15)	(58, 1.35)	(73, 1.18)	(85, 1.20)	6.42%
	(92, 1.88)	(107, 1.74)	(115, 1.69)	(123, 1.21)	(135, 1.55)	(144, 1.37)
	(152, 1.33)	(160, 1.36)	(165, 1.37)	(169, 0.82)
L18-B	(8, 4.37)	(15, 3.84)	(20, 4.12)	(26, 3.32)	(32, 2.59)	(48, 3.47)	14.09%
	(53, 3.10)	(65, 4.37)	(73, 4.42)	(87, 2.48)	(109, 2.52)	(112, 4.22)
	(145, 2.54)	(157, 1.72)	(168, 3.11)
L21-A	(16, 4.49)	(28, 2.28)	(45, 1.06)	(58, 2.30)	(72, 2.16)	(92, 1.91)	10.77%
	(98, 1.43)	(104, 0.78)	(110, 1.60)	(115, 2.09)	(154, 1.61)	(168, 1.91)

The corrosion rate is highly dependent on ambient environments. In this article, the corrosion current density of i_corr = 2.586 μA/cm² is selected as the typical working condition for de-icing salt (Duracrete, 2000). The initial parameters affecting corrosion, such as the surface chloride content (C_s), the apparent diffusion coefficient (D_app), and the critical chloride content (C_cr), are described as random variables, and their mean and variance are found in Val and Stewart (2003). In the analysis of the one-dimensional (1D) random fields, the midpoint method is adopted, and the element size of 0.5 m is used to discrete the three corroded beams from the left side of A web of each beam, that is, each beam is divided into 16 small elements, as shown in Figure 4. Once the beam element is dispersed, and the θ values of all parameters are known, the variation laws of the parameters along the beam length are also obtained by transforming the randomly generated data into the spatially correlated data based on MATLAB program. The fluctuation intensity of f_c corresponding to the different θ values are shown in Figure 5.

Figure 4.

One-dimensional random field.

Figure 5.

Fluctuation intensity of f_c in the X axis corresponding to the different θ values calculated by three methods: (a) Beam P1, (b) Beam P2, and (c) Beam P3.

As can be seen from Figure 5, when θ = 0, it means that f_c is a spatial random variable, but there is no correlation between the generated data of f_c along the beam. At this moment, f_c is generated randomly based on the specified probability density function without converting the randomly generated data into a spatially correlated data, hence f_c fluctuates rapidly and randomly along the beam at present. With the increase in θ value (compare the results of three estimation method: θ value obtained by the ACF method is the smallest, followed by the θ value obtained by the SVF method, and the θ value obtained by the BIC method is the largest), more smoother and fewer mutations curves of f_c along the beam were produced. When the spatial variability of f_c is not considered, it can be assumed that the θ value of f_c tends to be positive infinity, and f_c does not fluctuate and remains constant along the beam. Therefore, an increase in the coefficient of fluctuation will result in a reduction of the fluctuation intensity of concrete compressive strength. When the correlation length d, which is derived from θ, is increased to be larger than or equal to the beam length, the spatial variability of f_c can be ignored. This means that the variation trend of concrete compressive strength along the beam length is more stable with the increase in θ value. Meanwhile, the concrete compressive strength of each discrete element can be obtained from Figure 5. Similarly, the values of other spatial parameters of each discrete element can also be obtained by transforming the randomly generated data into the spatially correlated data. The bending moment of the corroded beam is calculated by converting the Π beam section to an equivalent T beam section in this article. Therefore, considering the influence of the spatial distributions of the concrete compressive strength, steel pitting factor, and other relative variables, the bending moments of all discrete elements can be calculated, and the minimum value of bending moment of all discrete elements is considered as the bending moment value of the whole beam. The theoretical values of the bending moment in the critical section (middle section or other section) of the three beams are shown in Table 3.

Results comparison

Because the failure mode of corroded beam in this experiment is bending failure, this article takes the pure bending of beam as the research object. In the four-point loading method, between the L/3 and 2L/3 section region of the tested beam is the pure bending region, and the corresponding discrete element is the RF element 6 to the RF element 11. Meanwhile, because the midpoint method is adopted in random field analysis, the bending moment value at the center of each discrete element is regarded as the bending moment value of the associated element. As such, the comparison of the measured bending moment of pure bending region and the theoretical value of corresponding discrete element can be conducted (see Figure 6).

Figure 6.

Comparison of measured value and theoretical value of bending moment: (a) Beam P1, (b) Beam P2, and (c) Beam P3.

Figure 6 shows that the theoretical bending moments of the corroded beams considering the estimation results of θ of all parameters obtained by BIC method have a better agreement with the experimental values as compared with those obtained from the other two methods. This indicates that the BIC method is more reasonable for estimating the scale of fluctuation of structural dimensions and material properties. It is also found from the Figure 6 that the critical section of the beam P1 is near the mid-span section (RF element 9 or RF element 8); however, the critical sections of beam P2 and beam P3 are located in RF element 7 and RF element 10, respectively. The load-carrying capacity test results show that the failure position of beam P1, beam P2, and beam P3 are in the range of (3.95 m, 4.20 m), (3.25 m, 3.55 m), and (4.65 m, 4.90 m), respectively, which are roughly consistent with the critical section position of the corresponding beams obtained by theoretical solution. The above analysis results not only show the validity of the BIC model in estimating the coefficient of structural parameters, but also confirm the importance of considering the spatial variability of structural parameters on the failure assessment of reinforced concrete structures. Meanwhile, it can be seen from Table 3 that the average relative error (ratio of the absolute error of two kinds of results and experimental value) of the theoretical value considering the estimation results obtained by BIC method is 3.76%, which is much less than the corresponding values considering the estimation results obtained by the ACF method (9.76%) and the SVF method (6.52%). Since the BIC method can make full use of the inspection information or the measured data, the result of the BIC method is more accurate as the relevant data increases. Therefore, the BIC method which is proposed in this article can be used to more reasonably estimate the scale of fluctuation in reliability assessment of reinforced concrete structures considering spatial variability.

Conclusion

A new method (BIC method) to estimate the scale of fluctuation of material and geometrical parameters based on the BIC is proposed, and the scale of fluctuation (θ) of concrete compressive strength (f_c) and steel pitting factor (R) are estimated by the BIC method based on the measured data. The scale of fluctuation of f_c and R predicted from the BIC method are compared with those obtained from the ACF method and SVF method. Then, the theoretical values of the bending moment of all beams are calculated considering the influence of the spatial distributions of the concrete compressive strength, pitting factor, and other relative variables. Finally, the theoretical results are compared with the experimental results of all beams. Therefore, the following conclusions are drawn:

The mean value of θ of concrete compressive strength obtained by the BIC method is 2.926, which is larger than that obtained by the ACF method and the SVF method. The fluctuation intensity of concrete compressive strength decreases with the increase in θ. When the correlation length d, which is derived from θ, is increased to be larger than or equal to the beam length, the spatial variability of f_c can be ignored.

Based on the measured value of R at different positions of 246 corroded steel bars, the mean and variance of the θ value of R are obtained by the three estimation methods, and the results show that there is no strong correlation between the spatial distribution of each pit on the surface of a corroded steel reinforcement.

The bending moments of the beams predicted from the BIC method are in better agreement with the experimental values than those predicted from the ACF method and the SVF method. The average relative error of the theoretical bending moment value of all beams based on the BIC method is 3.76%, which is much less than the corresponding value based on the ACF method (9.76%) and the SVF method (6.52%). This demonstrates that the proposed method in this article is more accurate to describe the spatial properties of material and geometrical parameters.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support of the National Basic Research Program of China (973 Program, Grant No. 2015CB057706), the National Natural Science Foundation of China (Grant Nos 51478050 and 51378081), the Natural Science Foundation of Hunan Province (Grant No. 17JJ486), the Excellent Young Research Program by the Department of Education at Hunan Province (15B015), and the Open Fund of Industry Key Laboratory of Traffic Infrastructure Security Risk Management (CSUST, Grant No. 16BCX08) are gratefully acknowledged.

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A new method for estimating the scale of fluctuation in reliability assessment of reinforced concrete structures considering spatial variability

Abstract

Keywords

Introduction

BIC method

Estimating the scale of fluctuation θ for fc

Structural description

Data acquisition

Estimated results of BIC method

Model verification

Load-carrying capacity test results

Theoretical analysis

Results comparison

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Estimating the scale of fluctuation θ for f_c