Advances in Structural Engineering

Deflection Reliability Analysis of PSC Box-Girder Bridge Under High-Speed Railway Loads by Tong Guo, Tie Liu and Aiqun Li

Reprinted from

Advances in Structural Engineering Volume 15 No. 11 2012

MULTI-SCIENCE PUBLISHING CO. LTD. 5 Wates Way, Brentwood, Essex CM15 9TB, United Kingdom

Deflection Reliability Analysis of PSC Box-Girder Bridge Under High-Speed Railway Loads Tong Guo*, Tie Liu and Aiqun Li Key Laboratory of Concrete and Prestressed Concrete Structures, Ministry of Education, Southeast University, Nanjing 210096, China

Abstract: A high-speed railway (HSR) construction boom has been observed in China during the past few years, in which prestressed concrete (PSC) box-girders are widely used for HSR bridges. Concerns have been raised regarding the long-term deflection of girders under the combined actions of concrete creep, shrinkage, and tendon relaxation, since the time-variant deflection significantly undermines the safety of high-speed trains. This paper presents the time-variant deflection reliability analysis of an existing HSR PSC box-girder bridge, in which a hybrid method, consisting of the response surface (RS) method, the finite element (FE) method and the checking point method (i.e., the JC method), is used. The pre-and post-cracking behaviors of the thinwalled box girder are described by using composite degenerated shell elements with the smeared cracking model and the Hordijk’s tension softening relation. In particular, the CEB-FIP model for creep and shrinkage is applied in the finite-element (FE) analyses. By using the probabilistic sensitivity analyses, random variables that significantly affect the time-variant deflection are selected for the reliability calculation. It is found that the deflection reliabilities are high in the early stage of bridge service but decrease rapidly during the first 10 to 20 service years. Higher speed limits result in a significant decrease in deflection reliability indices, which may fall below the target value prior to the expected service life. The proposed methodologies can be used in the design optimization, speed control and making rational maintenance or repair strategies for HSR PSC bridges.

Key words: deflection reliability, prestressed concrete box-girder, high-speed railway, creep and shrinkage, finite element modeling, response surface method.

1. INTRODUCTION A high-speed railway (HSR) construction boom has been undergoing in China during the past few years. Up to June 2011, the world’s longest high-speed rail (HSR) network with about 9,676 km of routes has been in service in China, including 3,515 km of rail lines with top speeds of 300 km/h. In these high-speed railways, the majority of the railway bridges adopt pre-stressed concrete (PSC) box-girder as their superstructures. Concerns have been raised regarding the long-term deflection of the PSC box-girders under the combined action of concrete shrinkage, creep and tendon

relaxation, because PSC box-girder bridges are structures that are particularly sensitive to their longtime deformations. Actual deflection after a longer service period, which was underestimated in initial analysis (Vitek 1997; Bazant et al. 2010), significantly influences the safety of passing vehicles or trains, especially when the speed is high. In the past decades, extensive research has been devoted to the time-variant performance of PSC boxgirder bridges; however, the progressive accumulation of deflection in PSC box-girder is a complicated mechanism, which involves many factors, such as

*Corresponding author. Email address: [email protected]; Fax: +86-25-83792076; Tel: +86-25-83790576.

Advances in Structural Engineering Vol. 15 No. 11 2012

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Deflection Reliability Analysis of PSC Box-Girder Bridge Under High-Speed Railway Loads

concrete creep, shrinkage, steel relaxation and corrosion, etc. For concrete creep alone, a number of predictive models and design codes have been proposed (ACI 1982; Bazant and Chern 1984; CEBFIP 1994; RILEM 1995; Gardner and Lockman 2001), in which parameters such as reinforcement ratio, posttensioning (PT) forces, concrete strength, ambient temperature, relative humidity, etc., are taken into account. Moreover, the box-girder has a closed crosssection, on which a unique shear lag effect (Kristek and Bazant 1987; Luo et al. 2002) exists. It could bring additional difficulties in accurate predicting the behaviors of bridges. In addition, the long-term behaviors of PSC bridges are highly uncertain phenomena, which should be dealt with by using the probabilistic approach. However, existing reliability analyses on PSC bridges were mainly focused on rectangular beams or T beams (Strauss et al. 2008; Stewart 2009; Wang and Qin 2007), and little has been done on the time-variant reliability of PSC box-girder bridges under highway or railway loads, which is a crucial step in the reliability-based design. In 2011, Guo et al. (2011a) proposed a probabilistic finite element approach for the time-dependent reliability analysis of PSC box-girders under highway loads, and factors including concrete creep, shrinkage and steel corrosion were considered. An adaptive importance sampling was later developed to increase the efficiency of reliability analysis (Guo et al. 2011b). These researches are extended to the deflection reliability analysis of high-speed railway PSC box-girder bridges as presented in this paper, and a hybrid method, consisting of the response surface (RS) method, the finite element (FE) method and the checking point method (i.e., the JC method) is used for reliability analyses. 2. FE MODELING METHOD FOR PSC BOXGIRDERS 2.1. Composite Degenerated Shell Element In this study, an eight-node composite degenerated shell element in the FE program DIANA (2008) is used, as shown in Fig. 1. This shell element is capable of modeling pre- and post-cracking behavior of thinwalled reinforced concrete structures and enables easy modeling of distributed reinforcement as well as prestressed tendons (Guo et al. 2011a). Non-prestressed reinforcement is modeled using a reinforcement grid, which is embedded in the shell element, as shown in Figure 1(a). The reinforcement ratios in two perpendicular directions are represented by two equivalent thicknesses teq (i.e the area of crosssection per unit length) of the grid, see Figure 1(b). Each

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(a)

(b)

Reinforcement Reinforcing grid bar

Y teq 1 (c)

Reinforcement grid ζ

X

t

Z η

(d)

Y

(e)

X

Y ξ

Integration point Location point Element node

s

X

Figure 1. Composite degenerated shell element

reinforcement grid is identified by the position parameter z, which defines the distance from this grid to the shell mid-surface, see Figure 1(c), and z should be no greater than the shell thickness t. The total area of the grid is divided into several segments, as shown in Figure 1(d), which contribute to the stiffness of the shell element. The location points, marked , define the positions of the segments in the shell element. An automatic tendon generation scheme included in DIANA (2008), is used, as shown in Figure 1(e). For this generation scheme, tendons are defined with a few location points and shape functions (i.e., straight, quadratic, or cubic curve). DIANA searches the shell elements for the intersections of the tendons with the shell element boundaries. These intersections serve as location points, which define the relationship of the tendons to their surrounding shell elements. The tendons are divided into segments by these location points, and numerical integration of each tendon segment is performed separately. 2.2. Concrete Creep and Shrinkage Goel et al. (2005) made a comparison between the existing models regarding concrete creep and shrinkage, including the double power law for creep (Bazant and Osman 1976), the ACI 209 model (ACI 1982), the double power logarithmic law for creep (Bazant and Chern 1984), the CEB-FIP model (CEBFIP 1994), the B3 model (RILEM 1995), and the GL2000 model (Gardner and Lockman 2001). It was concluded that the recently developed GL2000 model, the CEB-FIP model code 1990, and the B3 model, etc., are more reliable than the earlier creep models. This is because these models are based on extensive


Tong Guo, Tie Liu and Aiqun Li

research as well as experimental results. Therefore, the CEB-FIP model is used in this study. Effectiveness of this model is validated by comparing the predicted response with existing test results (Guo et al. 2011). In the CEB-FIP model, the evolution of concrete creep is mainly described via the creep function J(t, t0), formulated as

where T(τ) is the temperature of concrete at τ days. The modulus of elasticity of concrete at t days can be estimated as Ec (t ) = βcc ( t ) Ec 28

(4)

The shrinkage strains εs(t, ts) at an age of t days is J ( t , t0 ) =

φ ( t , t0 ) 1 + Ec (t0 ) Ec 28

(1)

where Ec (t0) is the modulus of elasticity at the concrete age of t0, and Ec28 corresponds to the value at the age of 28 days. φ(t, t0) is the creep coefficient, which is determined from the following hyperbolic power function:  1 − RH / RH 0  φ ( t , t0 ) = 1 + 1/ 3   0.46(h / 100)      t − t0 5.3 1    0.1 f   1/ 5  cm 28   0.1 + t0   β H + ( t − t0 ) 

0.3

(2)

where RH is the relative environmental humidity, RH0 equals to 100%, and h is the nominal size of the concrete member (mm), defined as 2Ac/u· Ac is the cross-sectional area and u is the perimeter in contact with the atmosphere. fcm28 stands for the mean compressive strength at the age of 28 days, and

(

β H = 150 1 + (1.2 RH )18

h + 250 ≤ 1500. ) 100

The evolution of concrete strength with time is described by Eqn 3(a). fcm (t ) = βcc ( t ) fcm 28

ε s ( t , ts ) = (160 + 10 β ( 9 − 0.1 fcm 28 )) t − ts

−6

× 10 β RH

(5a)

350 ( h / 100 )2 + ( t − ts )

where β is a shrinkage coefficient dependent on cement type, ts is the age of concrete at the beginning of shrinkage (day), and βRH is related to the environmental humidity RH as follows

)

(

3 −1.55 1 − ( RH / 100%) 40% ≤ RH ≤ 99% β RH =  RH > 99%  0.5

(5b)

2.3. Stress Relaxation Similar to the modeling of creep, stress relaxation is modeled via the relaxation function by using a generalized Maxwell model, which can be physically interpreted as a parallel set of springs and dampers (i.e., a set of Maxwell elements) as shown in Figure 2, where Ei and ηi denotes the stiffness of the spring and the damper, respectively. The stress-strain relationship for the generalized Maxwell model is expressed as

σ (t ) = ∫

(3a)

t −∞

E ( t , τ ) ε dτ

(6a)

where

((

βcc ( t ) = exp s 1 − 28 / teq

))

(3b)

is a time-variant coefficient and s takes the values 0.20, 0.25 and 0.38 for rapid hardening high strength cement, normal and rapid hardening cement, and slowly hardening cement, respectively. The equivalent age of concrete teq is defined as

E1

E2

En η0

E0 η

1

η

2

ηn

Maxwell element

t

teq = ∫ 4000 (1 / 273 − 1 / T (τ )) dτ 0


(3c)

Figure 2. Generalized maxwell model

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where E(t, τ) denotes the relaxation function, the relaxation function E(t, τ) can be expanded into a Dirichlet series n

E ( t , τ ) = ∑ Ei (τ ) e

−

t −τ λi

(6b)

σcr ft

I

G /h f

i=0

ε ult cr

where Ei(τ) is the time-variant stiffness of the Maxwell element in Figure 1, and the relaxation time for the Maxwell element is

λi = ηi / Ei

(6c)

Substituting Eqn 6(c) into Eqn 6(a) for time t and time (t + ∆t) respectively, and after summation, integration and subtraction, the stress increment in ∆t is ∆t   −   E t * λi D ∆ε − σ i ( t ) ∆σ = ∑  1 − e λi      ∆t i=0   n

( )

(7)

where t* is a sampling point at the midpoint the time increment, namely t + ∆t/2. More details regarding the generalized Maxwell model and the relaxation function can be found in Bazant and Wu (1974). The above concrete creep, shrinkage and steel relaxation models are integrated into the FE program DIANA. In this way, the influences of structural details (i.e., reinforcement arrangement, time-variant prestress level, etc.) as well as material properties (i.e., concrete strength and elastic modulus, etc.) can be considered automatically through FE analyses. 2.4. Crack Model and Tension Softening Relation A smeared cracking model with strain decomposition (De Borst 1987) and a tension softening relation of concrete (Hordijk et al. 1991), are used in this study. Figure 3 illustrates the relationship between crack stress σcr and crack strain εcr, which can be defined by

Figure 3. Nonlinear tension softening

The effectiveness of the above FE modeling approach has been validated based on existing test results (Guo et al. 2011a, b) 3. HYBRID RELIABILITY ANALYSIS METHOD Although the incorporation of finite element analyses (FEA) may increase the accuracy, as compared with explicit formula-based methods, the probabilistic nonlinear FEA are usually time-consuming. In this study, a hybrid method for reliability analyses is used, which consists of the RS method, the FE method and the JC method. Advantages of the three methods are combined to keep a balance between efficiency and accuracy. A similar hybrid method was proposed and validated by comparison with results of the direct Monte Carlo simulation (Cheng and Xiao 2005). The basic idea of the RS method is to represent the actual structural response by using a suitable approximation function (Nowak and Cho 2007; Ghosn et al. 2010), and one crucial step in the RS method is the design of experiments, which determines the design points such that the space of random input variables is explored in a most efficient way. In this study, the central composite design (CCD) (Montgomery 1991) is used. In the RS method, the structural response under a certain input can be approximated by the following fully quadratic polynomial: Z = g(X) ≈ Zr = gˆ (X) n

σ cr

(8)

ult is assumed as where the ultimate crack strain ε cr I I 5.136G f /(hft), G f is the Mode-I fracture energy and h is the equivalent crack length. The reduced tensile strength

ft equals to 0.739G If E / h .

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n

= a + ∑ bi X i + ∑ ci X i2 + i =1

   ε 3    ult  ft 1 + 3 cr  exp −6.93 ε cr − 0.027 ε cr 0 < ε cr < ε cr     ult   ult  ult ε cr  ε cr  =    ε cr    ult  0 ε cr > ε cr

εcr

i =1

∑

dij X i X j

(9)

1≤ i < j ≤ n

where n is the number of random variables, a, bi, ci and dij are the unknown coefficients to be determined, and the total number of these coefficients is [n(n + 1)/2] + n + 1. According to Das and Zheng (2000), in most cases higher order cross-correlation in Eqn 9 only has a small contribution to the accuracy of the analysis, but will significantly increase the required number of samplings of the response surface method. To keep a balance between the accuracy and efficiency, the cross-terms in



Eqn 9 are omitted so that the number of unknown coefficients is reduced to (2n + 1). To obtain the response surface and the structural reliabilities, an iteration procedure is proposed as shown in the following steps: (1) The coordinates of the initial iteration point, x = (x1, x2,···, xn)T, take the mean values of random variables µX. (2) (2n + 1) design points are generated by using the CCD with coordinates xi = µXi ± fσXi, (i = 1, ···, n), where f is a random factor (i.e., f = 2.0). For example, Figure 4 illustrates the five design points generated when there are two random variables (i.e., X1 and X2). (3) (2n + 1) times of FE analyses are made at these (2n + 1) design points respectively, so as to obtain the values of gˆi, (i = 1, 2, ···, 2n + 1); thereafter, the following linear equations set is formed: Aλ = gˆ

(10) T

where λ = ( a; b1 , b2 ,L , bn ; c1 , c2 ,L , cn ) , gˆi = ( gˆ1 , gˆ 2 ,L , gˆ 2 n +1 )T and

1   1 A=   M 1 

[ xi ]1× n [ xi ]1× n M

[ xi ]1× n

 xi2    1× n   xi2    1× n   M   xi2    1× n 

(11)

  0 0 0   2 2     +  0 − fdiiag σ Xi  diag  f σ Xi − 2 fxiσ Xi     2 2  0 fdiag σ Xi  diag  f σ Xi + 2 fxiσ Xi   X2 ( µx1, µx 2 + f σ x 2)

( µx 1, µx 2)

( µx 1, + f σ x 1, µ x 2)

X1 (µx1 − f σ x 1, µ x 2) (µx 1, µ x 2 − f σ x 2)

Figure 4. Design points in the X1 − X2 plane


Note that in Eqn 11, “diag” denotes the MATLAB function which returns a diagonal matrix. (4) The coefficients a, bi, ci (i = 1, 2,···, n) are determined by solving Eqn 10. (5) New design points x* are calculated by using the JC method, and FE analyses are made to obtain the g(x ˆ *). (6) According to g(x ˆ *), the design points for the next iteration are determined by using interpolation: x=µ X +

g( µ X ) n x∗ − µ X g( µ X ) − g( x ∗ )

(

)

(12)

(7) Steps 2, 3, 4, 5 and 6 are repeated until the difference between two iterations, X , is no greater than the predefined tolerance ε. (8) To this end, the reliability index β is calculated by using the obtained response surface and the JC method. The above reliability analysis is performed in the MATLAB environment and a code is developed to call the FE program DIANA. 4. DEFLECTION RELIABILITY OF PSC BOXGIRDER BRIDGE 4.1. Bridge Description The above hybrid reliability analysis method is illustrated on a double-cell PSC box-girder bridge in China, which has a span length of 23.5 m, an overall width of 24.4 m and a constant depth of 2.7 m, as shown in Figure 5(a). The bridge is not located in a chloride environment, so that corrosion is not considered in this analysis. The girder is simply supported at two ends, and Figures 5(b) and 5(c) shows the distributed reinforcements and the tendon arrangement, respectively, where each tendon has a diameter of 15.24 mm, corresponding to a crosssectional area of 140 mm2. The jacking stress σcon is 1,320 MPa, which is 71% of the standard ultimate strength (i.e., 1, 860 MPa) of the tendons. The design top speed of this HRS bridge is 250 km/h, but the speed limit is later increased to 300 km/h. 4.2. Load Cases Apart from the dead weight of the girder that is taken into account automatically by the FE program, the additional dead weights after the girder is erected (i.e., weight of rails, pavements, etc.) are estimated as 9.59 kN/m2, which is applied on the top plate of the girder. The live loads consist of three concentrated loads with a magnitude of 200 kN superimposed with a uniformly distributed load of 64 kN/m, as shown in Figure 6 (TB 10621-2009 2009). The longitudinal

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Mid–span section

1522

250

500

271324

Inclined slab

700

2926 6500/2

2700

120

Bottom plate 240

1959 2850

Proof wall

Support section 490

Mid slab

240

620

12200/2 500100150 1650 2600

1100

290

Top plate

1100

229

12200/2 2600

948

100 1650 150 500

450271

5600/2 6500/2

1959

620

2850

(a) Cross-section (dimension in cm) Support section

Mid–span section 4

2

2

7

4

5

1

2

1 1 φ 10

5 φ 18

2 φ 12

6 φ 20

3 φ 14

7 φ 22

4 φ 16

8 φ 25

5

1

6

4

4

1

2

8 6

2 3

1

2

4

4

2

(b) Distributed reinforcements

Mid–span section

Support section

(c) Tendon arrangement

Figure 5. PSC box-girder of a HSR bridge

3 × 200 kN 64 kN/m

64 kN/m 0.8 m 1.6 m 1.6 m 0.8 m

Figure 6. Design loads of HSR bridges

position of the live loads is determined from an influence line analysis to determine the worst case scenario (i.e., the maximum deflection). 4.3. Modeling of Uncertainties The random variables considered in the study are summarized in Table 1, in which the dead weight of the girder is described by a normal distribution with a mean value µ of 1.02 and a coefficient of variation (COV) of 0.022 (Li 1993), whereas the dead weights of subsidiary parts (i.e., rails, pavements, etc) follow a normal distribution with µ = 1.321 and COV = 0.068. The bias

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factor of live loads is described by an extreme distributed model with µ = 1.019 and COV = 0.042. A normal distributed dynamic load amplification (DLA) factor is applied to the static train load with a mean value of 1.366 and a COV of 0.04. Concrete strength is modeled by a normal distribution with a mean value of 56.50 MPa and a COV of 0.094 (GBT50283-1999 1999). According to the statistics provided by the China Meteorological Administration, the annual average temperature and ambient humidity of the bridge site are described by normal distributions with parameters listed in Table 1. The bias factors of the elastic modulus of distributed reinforcements, tendons and concrete are assumed to follow normal distributions, as shown in Table 1 (GB50153-92 1992). As the main resource of bridge bending capacity, PT forces play a significant role in the deflection-resistance, and its variance is modeled by a normal distribution with µ = 1.0 and COV = 0.04 (AL-Harthy and Frangopol 1994).



Table 1. Random variables used in this study Variable DW1 DW2 LL DLA CS (MPa)

Properties

Mean

COV

Distribution

Source

Bias factor of dead weight of box-girder Bias factor of dead weights of subsidiary parts Bias factor of live loads Dynamic load amplification

1.02

0.022

Normal (truncated at 0)

1.321

0.068


1.019 1.366

0.042 0.040

Extreme value distribution Normal (truncated at 1)

Li 1993

GBT50283-1999 1999 China Meteorological Administration China Meteorological Administration

Concrete strength

56.5

0.094


RH (%)

Annual relative humidity

77.1

0.039


T (°C)

Annual average temperature

15.4

0.044

Normal

1.0

0.060


GB50153-92

1.0

0.060


1992

1.0 1.0

0.060 0.040

Normal (truncated at 0) Normal (truncated at 0)

ES ET EC PTF

Bias factor of elastic modulus of distributed reinforcements Bias factor of elastic modulus of steel tendons Bias factor of elastic modulus of concrete Bias factor of PT forces

All the distributions of the random variables are truncated at certain values to avoid erroneous sampling in the following probabilistic analyses. In addition, the correlation between variables is not considered in this study. 4.4. FE Model of PSC Box-Girder In this study, one span of the bridge is modeled, and Figure 7(a) illustrates the finite element model mesh, where distributed reinforcement are simulated by a series of rebar grids as shown in Figure 7(b), and Figure 7(c) shows the embedded PT tendons. Initial material properties of concrete and steel are defined by using their standard properties. For concrete, a density of 2,400 kg/m3, a modulus of elasticity of 34,500 MPa, a Poisson’s ratio of 0.17 and the tension softening curve, as shown in Figure 3, are used. For steel reinforcing bars, a density of 7,850 kg/m3, a modulus of elasticity of 200,000 MPa, a Poisson’s ratio of 0.3 and a bilinear kinematic hardening model with a yield strength of 335 MPa are used. The Newton-Raphson iteration is used for the nonlinear analyses, in which the convergence criterion is a combination of energy and displacement with the convergence tolerance of 0.005 and 0.0005, respectively. Considering that there is a gap in the concrete proof wall in Figure 5(a) at a distance of 2 m, the proof wall does not contribute to the structural stiffness, and is only considered as distributed dead weights in the FE model.


AL-Harthy and Frangopol 1994

4.5. Probabilistic Sensitivity of Variables To get a better understanding on the influence of various random variables on bridge deflections, sensitivity analysis is made prior to reliability analyses. In contrast to the conventional deterministic sensitivity analysis, which uses a finite-difference scheme (that accounts for one variable at a time while keeping all others fixed), the present analysis is a probabilistic one, in which not only the slope at a particular location but also the random output parameters are considered. This is realized by using the Spearman rank-order correlation coefficient between input and output parameters, which measures a degree to which two variables are related (Sheskin 1997), as follows: n

∑ ( Ri − R ) ( Si − S ) rs =

i

n

2

n

∑ ( Ri − R ) ∑ ( Si − S ) i

2

( −1 ≤ rs ≤ 1)

(13)

i

where Ri is the rank of xi within the sample set [x1 x2 ···xn]T, Si is the rank of yi within the sample set – – [y1 y2 ···yn]T. R and S denote average ranks of a Ri and Si respectively. rs varies from −1.0 to 1.0. For rs = 0, the input and output parameters are fully independent, while for rs = 1, the input and output parameters are directly related. Thus, the correlation coefficient is used as a sensitivity measure of output parameters (i.e., deflections

2007


To perform the sensitivity analysis, 5,000 times of samples of variables are generated by using the truncated Latin Hypercube sampling (Guo et al. 2011b), and in each sampling, there are eleven random values corresponding to the eleven variables in Table 2. Correlation coefficients of variables at some selected service times are calculated according to Eqns 13 and 14, as listed in Table 2. Trial analyses by using 1,000, 2,000, 2,500 samples reveal that the sensitivity analyses results become unchanged when the number of samples is over 2,500; in other words, 2,500 times of sampling are enough for this sensitivity analysis. As can be observed from Table 2, when the jacking has just been finished, PTF, LL and EC are the three most significant factors affecting the girder deflection. Other variables, such as DW2 and DLA also have sensitivities greater than 0.1. However, the sensitivities of these variables decrease with time, indicating that as time goes on, the girder deflection becomes less sensitive to these variables. The sensitivity of concrete strength, CS, however, gradually increases with time, showing that concrete strength may has a long-term influence on the girder deflection. As to the relative humidity RH, the sensitivity reaches the peak value at about 5 years after the jacking and decreases thereafter. In addition, ES, ET, and T are regarded as insignificant variables, because they contribute little to the change in deflection; therefore, they are regarded as constants in the following reliability analyses, so as to reduce the required number of samples. It is worth noting that although RH and T are regarded as crucial factors related to concrete creep and shrinkage, in this particular case, the annual variances in the two variables are small and the mean humidity at the bridge site is high. As a result, the influence of RH and T on bridge deflection is not significant, as compared to other variables. For bridge sites with drier climates or large variance in ambient temperatures, the sensitivity analysis results can be different (Gardner and Lockman 2001).

(a) FE mesh

(b) Distributed reinforcements

(c) PT tendons

Figure 7. FE model of PSC box girder

in this study) with respect to input parameters (i.e., live loads, concrete strength, etc.). To facilitate the comparison of variable sensitivities, the normalized sensitivities are usually adopted, as shown in Eqn 14: rs , n ( xi ) =

rsabs , n ( xi ) m

∑

(14)

rsabs , n (x j )

j =1

where the superscript “abs” denotes the absolute value.

2008

4.6. Reliability Analyses Results Fig. 8 illustrates a typical deformed shape of the boxgirder at the mid-span, where it is observed that the girder deforms both in the longitudinal and transversal directions; therefore, the deflection is in fact the sum of the longitudinal and transversal deflections. This again shows the advantage of implementing the FE method in reliability analyses, because the transversal deflection is difficult to be accurately calculated from explicit equations, and therefore it has seldom been addressed in deflection reliability analyses. According to Figure 8, it can be observed that the locations between the mid slab and the inclined slab have the largest deflections; therefore, deflections of these points are taken for reliability analyses.



Table 2. Normalized variable sensitivities Time after jacking Variable DW1 DW2 LL DLA PTF ES ET EC CS RH T

Shortly after jacking

7 days

30 days

1 year

5 years

10 years

50 years

100 years

0.060 0.140 0.223 0.157 –0.238 –0.003 0.001 –0.162 –0.003 –0.002 –0.010

0.056 0.132 0.208 0.147 –0.220 –0.005 0.003 –0.152 –0.035 –0.040 –0.002

0.054 0.128 0.201 0.143 –0.211 –0.006 0.003 –0.148 –0.046 –0.059 0.001

0.051 0.122 0.192 0.137 –0.200 –0.008 0.005 –0.143 –0.066 –0.072 0.005

0.050 0.120 0.187 0.133 –0.194 –0.010 0.008 –0.141 –0.076 –0.076 0.006

0.050 0.120 0.188 0.134 –0.194 –0.010 0.009 –0.142 –0.080 –0.068 0.005

0.049 0.119 0.186 0.133 –0.192 –0.011 0.010 –0.142 –0.085 –0.070 0.004

0.049 0.119 0.186 0.132 –0.191 –0.011 0.010 –0.142 –0.086 –0.071 0.003

Locations with largest deformation

7

350 km/h 300 km/h 250 km/h

Undeformed shape

Deformed shape

Reliability index

6 5 4 3

Figure 8. Deformed shape of the mid-span cross-section β target

2 0

According to the Chinese code TB 10621-2009 (2009), the deflection limit for HSR bridges with spans no longer than 40 m is L/1400, L/1500, and L/1600 corresponding to the speed limit of 250 km/h, 300 km/h, and 350 km/h, respectively, where L is the bridge span. Based on the three deflection limits, deflection ultimate limit-states are defined. By using the proposed methodologies in Section 3, deflection reliabilities are calculated at various service years, so that the evolution of the reliability indices is obtained, as shown in Figure 9. It is found that the bridge has high reliabilities at the beginning of service, which is 7.2, 6.5 and 6.0 corresponding to the speed limit of 250 km/h, 300 km/h and 350 km/h, respectively. However, a rapid decrease in reliability indices is observed during the first 10 to 20 service years. Thereafter, the decrease becomes mild. This deterioration pattern is in accordance with the evolution of concrete creep, shrinkage and tendon relaxation, etc. When the speed limit is higher (i.e., 350 km/h), corresponding to a more strict deflection limit (i.e., L/1600), the deflection reliabilities become significantly smaller. Taking the bridge with a speed limit of 350 km/h for example, the reliability index fall below the target value of 2.3 (ISO 2394 1996) after about 28 years of service, indicating that the bridge may exhibit larger deflection than what


10

20

30

40 50 60 70 Time (year)

80

90 100

Figure 9. Time-variant deflection reliability

is expected; therefore, design optimization, maintenance or speed control actions may be needed. 5. CONCLUSIONS This paper presents the deflection reliability of a PSC box-girder bridge under high-speed railway loads. Subject to the limitations of the assumptions and parameters adopted in the study, the following conclusions are obtained: (1) The proposed FE modeling approach and the hybrid reliability method offer an effective tool for the deflection reliability analysis of HSR PSC box-girder bridges, which have unique structural behaviors, complex material constitutive relation and time-variant deterioration. The presented methodologies can be used in the design optimization, speed control and making rational maintenance or repair strategies for such bridges. (2) The probabilistic sensitivity analysis shows that the PT force, the live load and the elastic modulus are the three most significant factors affecting the girder deflection, and variable

2009


sensitivities vary with time. It is worth noting that this analysis, however, is site-specific. For bridges in drier regions or regions with larger variance in climates, the sensitivity analysis results may be different, and variables, such as relative humidity and ambient temperature, etc., may become more dominating factors. (3) Although the bridge has high deflection reliability in the early stage of service, the reliability index decreases rapidly due to creep, shrinkage and tendon relaxation, etc. Such decrease is most significant during the first 10 to 20 service years. For the bridge with the design top speed (i.e., 250 km/h), the reliability indices are higher than the target value although the service time; however, if the top speed is raised to 350 km/h, the reliability indices may fall below the target value prior to the expected service life. (4) Corrosion is not taken into account in this study due to the fact that the prototype HSR PSC bridge presented in Section 4 is not in a chloride environment. However, the proposed analytical model can be combined with a corrosion model (Guo et al. 2011a) to account for the influence of time-dependent corrosion on the deflection reliability when necessary. (5) Although emphasis in this study is placed on the deflection reliability, which is one of the control criteria for HSR bridges, the reliability in the ultimate limit state can be calculated in a similar way. The proposed methodologies also enable immediate applications to other PSC structures as well as other probabilistic problems regarding their time-variant reliabilities. ACKNOWLEDGEMENTS The support from: (1) the Program for New Century Excellent Talents in University under Grant No. NCET10-0326; (2) the Natural Science Foundation of Jiangsu (No. BC201120703); (3) the Fok Ying Tung Education Foundation (No.131074); and (4) a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions is gratefully acknowledged. The opinions and conclusions presented in this paper are those of the writers and do not necessarily reflect the views of the sponsoring organizations. REFERENCES AL-Harthy, A.S. and Frangopol, D.M. (1994). “Realiability-based design of prestressed concrete beams “, Journal of Structural Engineering, ASCE, Vol. 120, No. 11, pp. 3156–3177.

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