Stochastic long-term behavior of a reinforced concrete arch bridge

Abstract

Beipanjiang Bridge, a railway-reinforced concrete arch bridge with span of 445 m, is located in Guizhou, China. The self-shored technology with a frame of concrete-filled steel tube is applied in this construction, in which the frame performs as the falsework for casting concrete into arch ring and also serves as the major reinforcement of the permanent structure. The time-variant behaviors of long-span concrete arch bridge may be sensitive to the creep and shrinkage of concrete, especially when the staged construction is applied. Owing to the random property of concrete creep and shrinkage, the stochastic analysis is necessary to be employed in order to give a more rational and objective evaluation about the structural long-term behaviors. A hybrid approach including Monte Carlo sampling based on response surface method is developed in this study, based on which the statistical results of long-term deflection and time-variant stress are discussed. Finally, the stochastic contributions of input variables to structural responses are investigated with the sensitivity analyses.

Keywords

arch concrete creep randomness shrinkage

Introduction

Generally, it is difficult, or even technically unfeasible, to construct a long-span concrete arch bridge only relying on a ground-supported falsework system, especially with the arch crossing over a deep river or valley. In China, as a viable solution to this problem, the self-shored system with a frame of concrete-filled steel tube is gaining ground in the construction of long-span concrete arch bridges, among which the Wanxian Yangtze river bridge completed in 1997 has a span of 420 m (Liu et al., 2002). The first step of this self-shored construction is to build a truss arch with steel tube using the traditional cantilever launching method. In view of the steel tube being light, a relative simple back-anchor system could be utilized to balance the weight of truss during the launching process, and the light weight of steel tube system also helps to save the lifting capacity of crane. Once the construction of steel tube is finished, the core concrete will be infilled into the tube for enhancing the structural strength, stiffness, and stability. The structural system mainly composed of concrete-filled steel tubes, usually called “stiffening frame,” which serves as falsework for the follow-up construction stages, based on which the arch ring could be gradually built up through enveloping the frame with concrete (Liu et al., 2002).

Beipanjiang Bridge, a railway concrete arch bridge with the record span of 445 m, is located in Guizhou, China, and the layout and major geometrical configuration are given in Figure 1. The self-shored technology with stiffening frame is adopted in construction, the procedure of which is summarized in Figure 2.

Figure 1.

Layout and geometrical configuration of Beipanjiang Bridge: (a) layout (dimensions in m) and (b) cross section of arch rib (dimensions in cm).

Figure 2.

Construction procedure of arch: (a) forming the truss arch with steel tube (cast of concrete do not exceed ~63 days), (b) infilling the core concrete and forming the stiffening frame (64–87 days), (c) enveloping concrete over the outside box (88–507 days), and (d) enveloping concrete over the inside box (508–783 days).

For the concrete arch bridge constructed by stiffening frame technology, it can be predicted that the long-term behaviors caused by creep and shrinkage may be considerably sophisticated. First, since the stiffening frame serves as the major reinforcement, creep and shrinkage of concrete may cause stress redistribution among steel tube, core concrete, and enveloping concrete. Moreover, since the arch structure system is formed gradually, the age of concrete casted in each construction step is quite different. Therefore, the sequence of construction will make significant influence on the long-term deflection and stress redistribution process. Meanwhile, it should also be noted that the time-variant properties of core concrete are quite distinct from those of enveloping concrete. The core concrete is sealed by steel tube, and no moisture migration occurs; thus, drying creep and shrinkage are considerably small and only basic creep needs to be concerned (American Concrete Institute, 2008; Bazant and Baweja, 2000; Gardner and Lockman, 2001). On the contrary, since the enveloping concrete outside of steel tube is exposed to atmosphere directly, basic creep, drying creep, and shrinkage should be fully taken into account for describing structural long-term behaviors.

On account of the random nature of concrete creep and shrinkage, the deterministic analysis cannot give a rational estimation about long-term behaviors of concrete structures in probabilistic-based meaning (Bazant, 2001). Therefore, for structures sensitive to creep and shrinkage, it is strongly recommended to take into account uncertainties of these structural behaviors in long-term analysis, on which the initial works were pioneered by Madsen and Bazant (1983) and Bazant and Liu (1985). In their studies, based on the investigation on the uncertain resources, a stochastic model for creep and shrinkage was developed and then the mean and variance of creep and shrinkage effects are computed by approaches of point estimation (Madsen and Bazant, 1983) and Latin hypercube sampling (LHS) (Bazant and Liu, 1985), respectively. Bazant and Kim (1989, 1991) investigated the random long-term deflection and internal forces of prestressed concrete (PSC) segmental box-girder bridges with LHS. Subsequently, Yang (2005, 2007), Pan et al. (2011), and Ma and Wang (2013) extended those previous research works into practical engineering application. On serviceability reliability analysis, Li and Melchers (1992) and Stewart (1996) analyzed the uncertainty of creep and shrinkage using Monte Carlo simulation (MCS). A step-by-step calculation procedure is set up to quantify these stochastic effects in long-term behaviors of composite beams (Fan et al., 2010). Guo et al. (2011, 2012) studied the time-variant reliability of the PSC box-girder bridge subjected to the combining action of creep, shrinkage, and corrosion.

In this article, utilizing MCS, a hybrid approach including response surface method (RSM) is introduced in the stochastic long-term analysis of Beipanjiang Bridge. Generally, direct Monte Carlo (MC) approach is almost infeasible when the structural response cannot be expressed explicitly and finite element analysis must be required in every sampling. RSM provides a versatile and effective way to simplify the implicit analysis by composing an explicitly expressed polynomial function for numerical computation (Huh and Haldar, 2001). Only several runs of structural analysis are necessary to form response surface, which saves a great deal of computational cost in stochastic analysis.

Creep and shrinkage analyses

Finite element formulation

A general finite element analysis program based on degenerated beam element (Xiang et al., 2005) is utilized in this article; as a key part of the whole framework, the degeneration simulation has been used well in the formation of shell and beam elements (Ahmad et al., 1970; Worsak, 1978). Since the displacement field of degenerated beam is described in the light of Timoshenko beam theory, the three-dimensional strain-displacement equations can be retained. The position of any point in the degenerated beam element illustrated in Figure 3 can be given as

\vec{X} = \sum_{i = 1}^{2} N_{i} (r) ({\vec{X}}_{i} + y^{'} {\vec{V}}_{2} + z^{'} {\vec{V}}_{3})

(1)

in which $\vec{X}$ is the coordinates of any point in the element, ${\vec{X}}_{i}$ is the coordinates of node $i (i = 1 - 2)$ . ${\vec{V}}_{1}$ , ${\vec{V}}_{2}$ , and ${\vec{V}}_{3}$ form the local orthogonal coordinate system shown in Figure 1. $y'$ and $z'$ are the local coordinates in ${\vec{V}}_{2}$ and ${\vec{V}}_{3}$ directions, respectively. The interpolation function $N_{i} (r)$ is written as

N_{i} (r) = \frac{1}{2} (1 + r_{i} r)

(2)

where $r_{i}$ is the isoparametric coordinate of node i.

Figure 3.

The degenerated beam element.

The local orthogonal coordinate system shown in Figure 3 can be constructed with the following way

{\vec{V}}_{3} = {\vec{V}}_{1} \times {\vec{V}}_{2}

(3)

in which ${\vec{V}}_{1}$ is expressed as

{\vec{V}}_{1} = {(\begin{matrix} \frac{x_{2} - x_{1}}{L} & \frac{y_{2} - y_{1}}{L} & \frac{z_{2} - z_{1}}{L} \end{matrix})}^{T}

(4)

where L is the length of element. Then, with the given ${\vec{V}}_{2}$ , ${\vec{V}}_{3}$ can be finally determined through equation (3).

The displacement field can be expressed in the form

\vec{U} = \sum_{i = 1}^{2} N_{i} (r) [{\vec{U}}_{i} + {\vec{U}}_{α} (y', z')]

(5)

where $\vec{U}$ is the displacement vector at any point in the element, ${\vec{U}}_{i}$ is the nodal displacement vector at node i, and ${\vec{U}}_{α} (y', z')$ is relative nodal displacement produced by normal rotations. Considering the hypothesis of plane section, ${\vec{U}}_{α} (y', z')$ can be expressed as

{\begin{matrix} u_{i} (α) \\ v_{i} (α) \\ w_{i} (α) \end{matrix}} = | \begin{matrix} {\vec{V}}_{1} & {\vec{V}}_{2} & {\vec{V}}_{3} \\ {α'}_{i 1} & {α'}_{i 2} & {α'}_{i 3} \\ 0 & y' & z' \end{matrix} |

(6)

where $α'_{i 1}$ , $α'_{i 2}$ , and $α'_{i 3}$ are the normal rotations about local axes of ${\vec{V}}_{1}$ , ${\vec{V}}_{2}$ , and ${\vec{V}}_{3}$ , respectively. Considering displacement transformation relation, equation (6) can be written in terms of the normal rotations about global axes as

{\begin{matrix} u_{i} (α) \\ v_{i} (α) \\ w_{i} (α) \end{matrix}} = [\begin{matrix} 0 & y^{'} n_{2} + z^{'} n_{3} & - y^{'} m_{2} - z^{'} m_{3} \\ - y^{'} n_{2} - z^{'} n_{3} & 0 & y^{'} l_{2} + z^{'} l_{3} \\ y^{'} m_{2} + z^{'} m_{3} & - y^{'} l_{2} - z^{'} l_{3} & 0 \end{matrix}] {\begin{matrix} α_{i 1} \\ α_{i 2} \\ α_{i 3} \end{matrix}}

(7)

in which $(\begin{matrix} l_{2} & m_{2} & n_{2} \end{matrix})^{T}$ and $(\begin{matrix} l_{3} & m_{3} & n_{3} \end{matrix})^{T}$ represent the direction cosine of ${\vec{V}}_{2}$ and ${\vec{V}}_{3}$ in global coordinate system, respectively. Then, equation (5) can be rewritten as

\vec{U} = \sum_{i = 1}^{2} N_{i} (r) ({\vec{U}}_{i} + Φ {\vec{α}}_{i})

(8)

where

Φ = [\begin{matrix} 0 & y' n_{2} + z' n_{3} & - y' m_{2} - z' m_{3} \\ - y' n_{2} - z' n_{3} & 0 & y' l_{2} + z' l_{3} \\ y' m_{2} + z' m_{3} & - y' l_{2} - z' l_{3} & 0 \end{matrix}]

(9)

and ${\vec{α}}_{i}$ is the normal rotation vector about global axes.

The strain tensor of any point in the global coordinate system can be expressed as

ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i})

(10)

From equations (8) and (10), the relation between the global strain and displacement is derived as

\vec{ε} = [B] \vec{U}

(11)

According to the stress–strain characteristic of beam, the strain vector in the local coordinate system shown in Figure 1 can be defined as

\vec{\bar{ε}} = {(\begin{matrix} ε_{11} & ε_{12} & ε_{13} & ε_{23} \end{matrix})}^{T}

(12)

The local strain vector $\vec{\bar{ε}}$ can be achieved using the second-order tensor transformation relation, that is

\vec{\bar{ε}} = [T] \vec{ε} = [T] [B] \vec{U} = [\bar{B}] \vec{U}

(13)

in which $[\bar{B}]$ is the local strain matrix.

The cross section of arch is gradually formed because of the staged construction process, so almost, respectively, the steel tube, core concrete, and enveloping concrete of the same section contribute their stiffness and strength in different construction steps. This process is quite complicated and may be extremely difficult to be simulated through a unique element. In this study, a simple but effective approach is developed; the key measure of the approach is to discretize the steel tube, core concrete, and enveloping concrete into individual elements in order to activate them independently according to the steps of construction. Additionally, the displacement compatibility among different elements in one section is achieved by sharing the same node.

To integrate the stiffness matrix and internal force of degenerated beam element, the piecewise integration method is usually utilized (Ahmad et al., 1970; Worsak, 1978). In this method, the element is subdivided into a series of pieces, and one integration point is allocated to each piece. The stress of the integration point will be computed according to one-dimensional constitutive model, with the aid of which the different material behaviors of steel and concrete can be simulated conveniently (Xiang et al., 2005). A general finite element analysis procedure is applied in the program (Xiang et al., 2005), and the flowchart of the procedure is illustrated in Figure 4.

Figure 4.

The flowchart of the finite element analysis procedure.

Age-adjusted effective modulus method

Since shrinkage of concrete is independent to stress history, simulation of shrinkage effect is comparably simple and is easy to follow the similar scheme of temperature effect analysis (Xiang et al., 2005). Exact calculation of creep effect could be more complicated and should involve solution of an integral or differential equation, which may be inconvenient for engineering application. Through transforming the integral or differential equation into an algebraic problem, age-adjusted effective modulus method, which is developed by Bazant (1972), provides a simple but effective way in numerical computation. In this method, the long-term strain caused by time-variant stress can be written as

ε (t, t_{0}) = \frac{1 + φ (t, t_{0})}{E (t_{0})} σ (t_{0}) + \int_{t_{0}}^{t} \frac{1 + φ (t, τ)}{E (τ)} d σ (τ)

(14)

where $ε (t, t_{0})$ represents the stress-induced strain during duration $t_{0} - t$ , $E (τ)$ is the elastic modulus of concrete at age $τ$ , and $φ (t, τ)$ is the creep coefficient. Let

Δ ε (t, t_{0}) = ε (t, t_{0}) - \frac{σ (t_{0})}{E (t_{0})}, Δ σ (t, t_{0}) = σ (t) - σ (t_{0})

(15)

Then, with mean value theorem of integral, equation (14) can be rewritten as

Δ σ (t, t_{0}) = E_{φ} (t, t_{0}) [Δ ε (t, t_{0}) - \frac{σ (t_{0})}{E (t_{0})} φ (t, t_{0})]

(16)

in which $E_{φ} (t, t_{0})$ is named as the age-adjusted effective modulus. Equation (16) gives an algebraic relation between increment stress and strain caused by creep, which can be solved conveniently with general finite element analysis program. Age-adjusted effective modulus $E_{φ} (t, t_{0})$ is given as

E_{φ} (t, t_{0}) = \frac{E (t_{0})}{1 + χ (t, t_{0}) φ (t, t_{0})}

(17)

where $χ (t, t_{0})$ is the aging coefficient (Bazant, 1972).

Figure 5 gives the flowchart, the effects caused by creep and shrinkage are computed with age-adjusted effective modulus, and this process is calculated automatically in finite element analysis program.

Figure 5.

The flowchart of age-adjusted effective modulus method.

Creep and shrinkage models

Several creep and shrinkage models, such as ACI 209 model (American Concrete Institute, 2008), CEB-FIP MC90 model (Comite Euro-International Du Beton, 1993), B3 model (Bazant and Baweja, 2000), and GL2000 model (Gardner and Lockman, 2001), are proposed in those previous research works. In this study, MC90 model and GL2000 model are chosen to simulate the creep and shrinkage of concrete.

In this article, only the most basic creep effect need to be concerned for core concrete. But for enveloping concrete, basic creep, drying creep, and shrinkage should be all involved in long-term analysis.

MC90 model

The compliance function $J (t, τ)$ , which represents the strain of concrete at age of t caused by a unit stress acted at age of $τ$ , is defined in MC90 model as

J (t, τ) = \frac{1}{E_{c} (τ)} + \frac{1}{E_{c}} φ (t, τ)

(18)

where $E_{c} (τ)$ and E_c are the elastic moduli of concrete at age of $τ$ and 28 days, respectively. $φ (t, τ)$ is the creep coefficient, which is given as

φ (t, τ) = ϕ_{R H} β (f_{c m}) β (τ) β_{c} (t - τ)

(19)

in which

\begin{matrix} ϕ_{R H} = 1 + \frac{1 - h}{0.46 {(\frac{0.02 A_{c}}{u})}^{\frac{1}{3}}}, β (f_{c m}) = \frac{5.3}{{(0.1 f_{c m})}^{0.5}}, \\ β (τ) = \frac{1}{0.1 + τ^{0.2}}, β_{c} (t - τ) = {[\frac{t - τ}{β_{H} + (t - τ)}]}^{0.3}, \\ β_{H} = 1.5 [1 + {(1.2 h)}^{18}] \frac{2 A_{c}}{u} + 250 \leq 1500 \end{matrix}

(20)

where A_c is the area of cross section (mm²), u is the perimeter of the member in contact with the atmosphere (mm), h is the ambient humidity expressed as a decimal, and f_cm is the mean compressive strength of concrete at age of 28 days (MPa). When only the basic creep is involved, the ambient humidity h in equation (20) can be set to be 1.0.

Prediction of shrinkage $ε_{s h}$ with MC90 model can be written as

ε_{s h} (t, t_{s}) = ε_{s} (f_{c m}) β_{R H} β_{s} (t - t_{s})

(21)

where

\begin{matrix} ε_{s} (f_{c m}) = [160 + 10 β_{s c} (9 - \frac{f_{c m}}{10})] \times 10^{- 6} \\ β_{R H} = {\begin{matrix} - 1.55 (1 - h^{3}) & 0.4 \leq h \leq 0.99 \\ 0.25 & h \geq 0.99 \end{matrix} \\ β_{s} (t - t_{s}) = {[\frac{t - t_{s}}{350 {(\frac{2 A_{c}}{100 μ})}^{2} + t - t_{s}}]}^{0.5} \end{matrix}

(22)

in which $β_{s c}$ is a parameter related to cement type (Comite Euro-International Du Beton, 1993) and $t_{s}$ is the age of shrinkage beginning.

GL2000 model

The compliance function of GL2000 model has the same form as that defined in equation (18), and the creep coefficient is expressed as

φ (t, τ) = Φ (t_{c}) {2 [\frac{{(t - τ)}^{0.3}}{{(t - τ)}^{0.3} + 14}] + \sqrt{\frac{7}{τ}} \sqrt{\frac{(t - τ)}{(t - τ) + 7}} + 2.5 (1 - 1.086 h^{2}) \sqrt{\frac{(t - τ)}{(t - τ) + 0.15 {(\frac{V}{S})}^{2}}}}

(23)

where $V / S$ represents the volume–surface ratio (mm), the first and second terms in brace of right hand represent basic creep, and the third term is corresponding to drying creep. $Φ (t_{c})$ is expressed as

Φ (t_{c}) = {\begin{matrix} 1 & τ \leq t_{c} \\ \sqrt{1 - \sqrt{\frac{(τ - t_{c})}{(τ - t_{c}) + 0.15 {(\frac{V}{S})}^{2}}}} & τ > t_{c} \end{matrix}

(24)

in which $t_{c}$ is the age of drying beginning. When only basic creep is concerned, $Φ (t_{c})$ is set to be 1.0.

Shrinkage can be predicted with the following equation in GL2000 model as

ε_{s h} (t) = ε_{s h u} β (h) β (t)

(25)

where

ε_{s h u} = 0.001 K \sqrt{\frac{30}{f_{c m}}}, β (h) = 1 - 1.18 h^{4}, β (t) = \sqrt{\frac{t - t_{c}}{t - t_{c} + 0.15 {(\frac{V}{S})}^{2}}}

(26)

in which K is a coefficient dependant on cement type (Gardner and Lockman, 2001).

Random analysis

Random variables

Although various resources may contribute to statistical scatter of creep and shrinkage effects, it is most crucial to take into full account the uncertainties of the analysis model. Generally, the stochastic model for the time-variant strain can be written as

ε (t, τ) = [α_{1} J (t, τ)] σ (τ) + \int_{τ}^{t} [α_{1} J (t, τ')] d σ (τ') + α_{2} ε_{s h}

(27)

in which $α_{1}$ and $α_{2}$ are the uncertain parameters related to the creep and shrinkage models, respectively, and $σ (\cdot)$ represents the time-variant stress. $α_{1}$ and $α_{2}$ can be assumed to be normally distributed, and the mean value and coefficient of variation (COV) are given as (Bazant and Baweja, 2000; Gardner, 2004)

\begin{matrix} E [α_{1}] = 1.0, C O V [α_{1}] = {\begin{matrix} 0.35 & MC 90 \\ 0.26 & GL 2000 \end{matrix} \\ E [α_{2}] = 1.0, C O V [α_{2}] = {\begin{matrix} 0.46 & MC 90 \\ 0.25 & GL 2000 \end{matrix} \end{matrix}

(28)

Two parameters, that is, the mean compressive strength of concrete $f_{c m}$ and the ambient humidity h, are taken as the random variables. The randomness of mean compressive strength of concrete is expressed as $α_{3} f_{c m}$ , in which $α_{3}$ is a random variable with distribution $N (1, 0.1)$ (Comite Euro-International Du Beton, 1993). Based on the meteorological records of the zone where Beipanjiang Bridge is located, the randomness of the ambient humidity can be expressed as $α_{4} h$ , in which $α_{4}$ is a uniform distribution variable with mean 1.0 and COV 0.039 (China Railway Eryuan Engineering Group Co. Ltd, 2010).

Furthermore, the randomness of the elastic modulus of concrete needs to be involved in the stochastic analysis. As shown in equation (1), for instantaneous elastic strain computation, the elastic modulus at age of $τ$ is used. Meanwhile, the elastic modulus at age of 28 days is adopted in the second term of left hand of equation (18). Generally, it is widely accepted that strong correlation likely exists between the concrete elastic modulus and compressive strength. By directly giving a correlation coefficient between them at age of 28 days, Keitel and Osburg (2010) involve the correlation in random analysis for creep and shrinkage effects.

An alternative approach comes into use in this article. Through multiplying the deterministic development function of the elastic modulus with a random variable, the stochastic correlations between the time-variant elastic modulus of concrete and its mean compressive strength at age of 28 days for MC90 model and GL2000 model are defined, respectively, as

\begin{matrix} E_{c} (t) = α_{5} \cdot 2.15 \times 10^{4} {(0.1 f_{c m})}^{\frac{1}{3}} \\ \sqrt{\exp {s (1 - \sqrt{\frac{28}{t}})}} forMC 90 model \\ E_{c} (t) = α_{5} \cdot (3500 + 4300 \sqrt{f_{c m} \frac{t^{0.75}}{a + b t^{0.75}}}) \\ for GL 2000 model \end{matrix}

(29)

where s, a, and b are constants related to cement type (Comite Euro-International Du Beton, 1993; Gardner, 2004), $α_{5}$ is an introduced random variable, and the function following $α_{5}$ is the deterministic model for development of the elastic modulus, given in MC90 and GL2000 model. Australia Standard 3600 (AS 3600-2001, 2001) states that the prediction value of elastic modulus may have ±20% variation, therefore, $α_{5}$ is assumed to be a normally distributed variable with mean 1.0 and COV 0.2 in this study.

Then, the stochastic model for long-term strain can be finally written as

\begin{matrix} ε (t, τ) = σ (τ) \cdot α_{1} [\frac{1}{α_{5} E_{c} (α_{3} f_{c m}, τ)} + \frac{φ (t, τ, α_{3} f_{c m}, α_{4} h)}{α_{5} E_{c} (α_{3} f_{c m})}] \\ + \int_{τ}^{t} α_{1} [\frac{1}{α_{5} E_{c} (α_{3} f_{c m}, τ')} + \frac{φ (t, τ', α_{3} f_{c m}, α_{4} h)}{α_{5} E_{c} (α_{3} f_{c m})}] \\ d σ (τ') + α_{2} ε_{s h} (t, α_{3} f_{c m}, α_{4} h) \end{matrix}

(30)

Moreover, multiplied with a random variable $α_{6}$ , the randomness of the magnitude of sustained load is included. A strict quality control about variation of sustained load is imposed on the construction of Beipanjiang Bridge, and $α_{6}$ is assumed to be normally distributed with mean 1.0 and COV 0.05 in this research.

Generally, the practical construction process cannot be perfectly consistent with the design procedure. In view of this kind of uncertainty, the time duration of the construction steps related to cast of concrete is noted as a random variable $α_{7}$ . It is required in the design document that the variation of time duration of concrete casting is not more than 1 day of the mean duration of 9 days, and then, $α_{7}$ is assumed to be uniformly distributed and COV equals 0.11. All the random variables involved in this study are listed in Table 1.

Table 1.

Random variables.

Random variables	Description	Mean	COV	Distribution
$α_{1}$	Uncertain parameter of creep	1.0	0.35(MC90) 0.26(GL2000)	Normal
$α_{2}$	Uncertain parameter of shrinkage	1.0	0.46(MC90) 0.25(GL2000)	Normal
$α_{3}$	Uncertain parameter of compressive strength	1.0	0.1	Normal
$α_{4}$	Uncertain parameter of ambient humidity	1.0	0.039	Uniform
$α_{5}$	Uncertain parameter of elastic modulus	1.0	0.2	Normal
$α_{6}$	Uncertain parameter of sustained load magnitude	1.0	0.05	Normal
$α_{7}$	Uncertain parameter of concrete age	0.0	0.11	Uniform

COV: coefficient of variation.

Sampling methods

LHS

LHS, an efficient algorithm to estimate the mean and variance of random responses, was first introduced into the stochastic analysis of creep and shrinkage effects of the concrete structures by Bazant and Liu (1985). Compared with the traditional MC sampling technology, LHS employs only 2N–3N times of sampling, in which N indicates the number of random variables.

Generally, two major steps are involved in LHS, that is, strategically sampling for every variable and randomly ordering for sampling results. And the flowchart of the procedure is illustrated in Figure 6. More details about this process can be referred to the works of Bazant and Liu (1985).

Figure 6.

The flowchart of Latin hypercube sampling.

MC sampling with RSM

RSM is a useful tool for random structural analysis, the basic idea of which is to approximately estimate the implicit function by an explicit function that is easier to deal with (Huh and Haldar, 2001). The form of response surface is a second-order polynomial without cross terms in this study

g (x) = a + \sum_{i = 1}^{N} b_{i} x_{i} + \sum_{i = 1}^{N} c_{i} x_{i}^{2}

(31)

where $g (x)$ represents the response surface function, $x_{i} (i = 1 - N)$ are the input variables, N is the total number of variables, and a, b_i, and c_i (i = 1–N) are the 2N + 1 unknown coefficients to be determined. To determine those coefficients a, b_i, and c_i, saturated design, in which the required number of experimental points (one central point and 2N star points) is the same as that of the unknown coefficients, is adopted. Since the probability distribution is what this research focuses on, the central point is selected to be the mean value of random variables, and 2N star points composed of one at +2 and one at −2 for each variable in the coded variable space. With the explicit second-order polynomial, MC sampling is utilized to obtain the stochastic responses. Only 2N + 1 times of the finite element analysis are involved in forming response surface, which decreases the cost of computation efficiently. Meanwhile, the probability distribution information of long-term behavior can be obtained with the same sampling method as well. The corresponding procedure is summarized in Figure 7.

Figure 7.

The flowchart of Monte Carlo sampling with response surface method.

Generally, finite element analysis method is used only to obtain the response result of an individual sample in LHS and MCS with RSM. Meanwhile, LHS is a special type of MCS, and the results obtained from LHS can agree with those from direct MCS. Meanwhile, MCS with RSM is introduced, and the sampling process is carried on an explicit function determined by response surface. To verify the accuracy, the results obtained from MCS with RSM are compared with those from LHS.

Numerical simulation of stochastic long-term behavior of Beipanjiang Bridge

Long-term deflection of crown

For the long-span railway concrete bridges, the time-variant deflection caused by creep and shrinkage of concrete makes negative impact on the smoothness of track profile, which may increase the vibrations of train and bridge. Therefore, to ensure the safety of high-speed train running, the studies on long-term deflection of railway concrete bridges are critical.

Since the deformation after opening to traffic is the major cause for deterioration of the track profile, the relative deflections referred to the completion configuration are mainly investigated. For simplicity, only the probability density functions of the relative deflection at crown based on MC90 model are plotted in Figure 8, and the relative deflection at crown is obtained using the general and versatile nonlinear finite element analysis program developed by authors, specifically discussed in the “Finite element formulation” section. To give a comparison between the results from LHS and MC with RSM, in the following figures, the results of $μ \pm 2 σ$ computed by LHS are presented, and the values of 2.28% and 97.72% fractiles are drawn from the probability distribution result obtained from MC combining with RSM. When the response is normal distribution, it can be expected that $μ \pm 2 σ$ will be corresponding to 2.28% and 97.72% fractiles.

Figure 8.

Probability density functions of long-term deflection at crown.

It can be viewed from Figure 8 that the mean and variance of deflection increase with time significantly. The distribution of long-term deflection is not assumed in prior, but as a result of calculation, it can be drawn from Figure 8 that the long-term deflection is normally distributed, and the values of $μ \pm 2 σ$ obtained from LHS agree well with 2.28% and 97.72% fractiles obtained from MC combining with RSM.

Time-variant stress

As mentioned above, obvious stress redistribution will occur among steel tube, core concrete, and enveloping concrete. The crown section is selected to give an illustration about this phenomenon, and the members of steel tube at bottom chord, core concrete at inside bottom chord, and enveloping concrete at bottom flange of outside box are mainly concerned. The stress redistribution process between the steel tube, the core concrete, and the enveloping concrete is shown in Figures 9 to 11. It can be observed in these figures that the stress of steel tube slowly increases over time, and the stress of the enveloping concrete decreases with time. Although the mean stress of core concrete almost keeps unchanging with time, Figure 10 shows that the randomness of stress of core concrete becomes scatter gradually. Moreover, the distribution of stress of core concrete fails to obey normal distribution, and an unsymmetrical distribution form can be observed. Therefore, obvious discrepancy can be viewed between $μ \pm 2 σ$ and 2.28% fractiles in upper and lower bound. So, in this case, it makes more sense to evaluate long-term behavior based on probability distribution.

Figure 9.

Probability density functions of stress of steel tube at bottom chord.

Figure 10.

Probability density functions of stress of core concrete at inside bottom chord.

Figure 11.

Probability density functions of stress of enveloping concrete at bottom flange of outside box.

Since the steel tube performs as a platform for the following construction procedure, the stress of steel tube is comparably high after completion of the arch structure. The potential risk of steel tube yield has been emphasized in the previous engineering application (Liu et al., 2002). Owing to various uncertain resources, it can be found from Figure 9 that the stress of steel tube exhibits considerable scatter. For example, over 10 years after completion of arch structures, the values of mean and 2.25% fractile of the stress of steel tube are −200 and −267 MPa (MC90 model), −218 and −281 MPa (GL2000 model), respectively. Thus, it can be seen from above results that the exceeding risk of stress could be enormously underestimated if based on the mean response get from deterministic analysis.

Sensitivity analysis

Generally, the sensitivity analysis is an effective way to measure the contribution of various random variables to stochastic responses. Based on the response surface, the sensitivity of structural response corresponding to each random variable is defined as

ϕ_{x i} = \frac{\frac{\partial g (x)}{\partial x_{i}} σ_{x_{i}} | x_{i} = u_{x_{i}}}{\sqrt{\sum_{j = 1}^{N} {(\frac{\partial g (x)}{\partial x_{j}} σ_{x_{j}} | x_{j} = u_{x_{j}})}^{2}}}

(32)

where $u_{x_{i}}$ and $σ_{x_{i}}$ represent the mean and standard deviation of random variable $x_{i}$ . The gradient of the structural response to the random variable is scaled by $σ_{x_{i}}$ to make a dimensionless measurement. Meanwhile, this definition likely provides a statistical sense which reflects the relative contribution of the uncertainty of each variable.

The sensitivity results of long-term deflection and stress of crown are plotted in Figures 12 to 14. From Figure 12, it can be observed that the most important resources of randomness for long-term deflection of crown are the model uncertainties of shrinkage and creep $(α_{2} and α_{1})$ . Moreover, the sensitivity of long-term deflection to the model uncertainty of shrinkage $(α_{2})$ increases with time while the contribution of the randomness of creep model $(α_{1})$ decreases gradually. If GL2000 model is adopted, it can be viewed that the ambient humidity $(α_{4})$ is one of the major uncertainties for the stochastic deflection. On the contrary, the influence of the ambient humidity $(α_{4})$ is comparably small when MC90 model is used.

Figure 12.

The sensitivity results of the deflection at crown: (a) CEB90 model and (b) GL2000 model.

Figure 13.

The sensitivity results of the stress of steel tube at bottom chord of crown: (a) CEB90 model and (b) GL2000 model.

Figure 14.

The sensitivity results of the stress of core concrete at inside bottom chord of crown: (a) CEB90 model and (b) GL2000 model.

Figures 13 to 15 show that randomness of creep model $(α_{1})$ is the most important uncertainty resource for the time-variant stress, and another major contribution to randomness is the model uncertainty of the elastic modulus of concrete $(α_{5})$ . As a result of utilizing GL2000, the contribution of uncertainty of the elastic modulus of concrete $(α_{5})$ is comparably greater than that of using MC90 model. Moreover, Figures 13 to 15 also show that the sensitivity of stress to the time duration of construction steps related to cast of concrete $(α_{7})$ is a trivial value that is ignorable. However, it is noteworthy that the random contribution of the variable to long-term deflection cannot be neglected as illustrated in Figure 12.

Figure 15.

The sensitivity results of the stress of enveloping concrete at bottom flange of outside box of crown: (a) CEB90 model and (b) GL2000 model.

Conclusion

To investigate the long-term stochastic behavior of a long-span concrete arch bridge, a hybrid approach combining MC sampling and RSM is developed, and the accuracy of the method is verified by comparing with the results of LHS.

The numerical results show the mean and variance of long-term deflection increase with time. To illustrate the random process of stress redistribution, the time-variant stress of steel tube in three different parts of a long-span concrete arch bridge is investigated, and the results reflect the distribution of time-variant stress is q divergent.

Especially, for stress of steel tube, the mean value from deterministic analysis may significantly underestimate the risk of stress exceeding. Therefore, random analysis should be strongly recommended to provide a probabilistic-based evaluation in cases like this.

Finally, the sensitivity analyses show that in addition to major resources as creep and shrinkage, the contributions from other factors, such as ambient humidity and elastic modulus of concrete, cannot be neglected.

It is undeniable that there are still some other time-variant mechanisms, such as fatigue of concrete or environmental-induced deteriorating probably need to be taken into account, but the proposed tools and framework provide a meaningful approach for further research.

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: This study was finically supported by the National Natural Science Foundation of China (Grant No. 51378432), the Science and Technology Program of Guizhou Provincial Transportation Department (Grant No. 2015-123-041), the Key Scientific Research Fund Project of Xihua University (Grant No. Z1510611), and the Open Research Fund of Xihua University Key Laboratory of Green Building and Energy Saving (Grant No. szjj2015-077). The support is sincerely appreciated.

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