Probabilistic Risk Assessment of a PSC Box Girder Railway Bridge System

Considering Construction Phases Using Six Sigma Methodology

1T. Cho1, 1J. Kim1, K. Lee, 1D. Cha1 Korea Rail Network Authority, Daejeon, Republic of Korea1

Abstract

Currently the prediction and the control of prestressing force and camber are designed deterministically in the design of cable stayed bridges or prestressed concrete bridges. However, the variation of qualities of materials and external loads have different types of probabilistic distributions, which could make additional error in the prediction and the control of errors. Therefore, the uncertainties in the resistance and loads should have to be considered in a probabilistic manner. To develop a probabilistic risk assessment technique in prestressed concrete box girder railway bridges, the important random variables are determined by a analytical hierarchy process (AHP) method, which are selected for the risk assessment of the target PSC box girder bridge constructed by a MSS method. The limit state functions are determined to investigate the risk of tensile cracks in upper and lower flange concrete, just after the moving of scaffolding, and the risks of the prestressing loss at each construction stage. For composing the implicit limit state function, Response Surface Method (RSM) is selected to evaluate the reliability of the implicit limit states of complex structures. The basic RSM could diverges depending on the nonlinearity of the limit states. For the improved convergence, iterations are performed to find the more probable failure points, which are closer to the limit states by updating design matrix, the input data to compose the response surfaces of considered systems. For maximizing the adaptation of RSM, a diagonal weighting matrix is used, which accelerates the convergence of reliability. Accordingly in the target PSC Railway Bridge, the linear adaptive weighted response surface method combined with advanced first order second moment method have been used for the evaluation of reliabilities of the considered limit states. Consequently, risk assessments have been performed for the limit states of the response of the target bridge, as a resistance term using the ultimate stress and prestress loss with the load term using the rupture stress and expected prestress loss, based on the Korean design specifications and ACI specification for each construction stages of PSC box girder bridges built by Movable Scaffolding Method.

Introduction and motivations

The processing time of MCS is approximately inversely proportional to the probability of failure. To overcome the drawbacks of MCS by reducing the variance of MCS, the stratified sampling in survey engineering [1], applied to MCS [2,3], Latin hyper cube method [4,5], and Markov chain modeling [6,7] have been developed. One of other approaches accounting for uncertainties is stochastic finite element method (SFEM), the modified FEM for spectral and stochastic random fields. Stochastic finite element method could solve the limitation of MCS by a perturbation technique [8,9] or by a weighted integral method [10,11] to incorporate uncertainty in the structural system. But it is limited to the specific program, in which the mean and coefficient of variation for random variables are programmed by a perturbation or by a weighted integral method. Therefore, it is not applicable when using commercial programs or any program that is not prepared the tasks. Subsequently, MCS, SFEM or basic RSM may not be applied to a real complex structure in a reasonable time of calculations or in an acceptable exactness. For simulating any system indirectly in reduced time and effort, the response surface method (RSM) was developed by Box and Wilson [12] to maximize chemical factory operation area, and it is now widely used in many areas in civil engineering fields [13-16]. Since it represents the

response of a complicate structure in an explicit form, it can be applied to a prediction of the behavior of structures, such as the prediction of deteriorated concrete structures by cyclic freeze-thaw [17], or the probabilistic prediction of failure modes of a bridge structure [18]. After approximating the structural response to a fitted explicit limit state function, the RSM is used to evaluate the probability of occurrence. Using the fitted function of selected important variables, the limit state functions are used for the safety evaluations, with much less numbers of structural analyses than that of MCS. Among many strategies and techniques in management system, a technique in decision making of analytical hierarchy process (AHP) [19] is adopted in the present study. In order to model and evaluate the uncertainties in resistance and load, the important design variables have been determined by employing AHP method. The following objectives are studied in this paper: 1) Decision making of important design variables, which compose response surface from the considered analysis program 2) Risk assessment for the considered limit states, composed by the developed LAW-RSM

Determination of important design random variables

In six sigma methodology, it is approached to obtain the final goal by the following order: 1) Define, 2) Measure, 3) Analyze, 4) Improve, and 5) Control, where the important causes or variables are determined at the forth step, Analyze. One of the most widely employed methods at the step is two dimensional comparison matrix method. Other methods are dotting in brain storming and pay off method, the matrix comparison method between cost and effect. For instance, two dimensional X-Y matrix comparison method is applied to determine the important design variables, shown in Table 1.

Input Variables (X's) Output Variables

(Y's)

External load at each

construction phase

Internal resistance capacity at each

construction phase

Internal stress at

each construc

tion phase

Rank % Rank

Output Ranking

2 2 1

Association Table

Prestressing

Prestressing force

2 2 2 14 0.09091

Diameter of tendons

1 2 2 12 0.07792

Curvature coefficient

2 1 1 9 0.05844

Wobble friction coefficient

2 1 1 9 0.05844

Eccentricity of tendons centroid of prestressing

tendons.

2 2 2 14 0.09091

Manufacturing company

2 1 1 9 0.05844

Concrete

Compressive strength,

1 2 2 12 0.07792

Creep coefficient 2 1 2 10 0.06494Shrinkage coefficient

2 1 2 10 0.06494

Elasticity of concrete.

1 2 2 12 0.07792

Relative humidity

1 1 1 14 0.09091

Regional values 1 2 2 12 0.07792Scaffol

ding Stiffness 2 1 2 10 0.06494Geometry 1 1 1 7 0.04545

Table 1 XY Matrix comparison method The input design variables are compared and evaluated for the load, resistance and stress during construction phases, followed by summing the points after multiplied with the associated points in column. Therefore, the rank is determined, as shown in last column in Table 1. The strong point of the X-Y matrix comparison method is simplicity and clarity, when dealing multiple decision criteria for multiple design variables. However, the evaluation not only subjective in nature, but no additional criteria is provided for the validation of the evaluated results as well. Therefore, as an advance method Analytic Hierarchy Process (AHP) was developed, which provides consistency during the evaluation processes with independent hierarchy process. The AHP, developed by Saaty [19], is a technique for considering data or information about a decision in a systematic manner. The AHP method mainly deals how to solve decision making problems, which involves uncertainties and multiple criteria characteristics. It is based on three principles: first, constructing the hierarchy; second, priority setting, and third, logical consistency.

Determining random variables of the target bridge based on the Analytic Hierarchy Process (AHP) method

The AHP method has the distinct advantages which can decompose a complex decision problem into sub-problems, build hierarchies of criteria, and determine the weights of criteria. A definite number within a 1–9 scale to the pair-wise comparison is evaluated so that the priority vector can be computed. Assume two factors of 1 and 2, if 1 and 2 are equally important, then it has a scale of 1; if 1 is weakly more important than 2, then it has a scale of 3; scales of 5, 7 and 9 are used to describe strongly more important, very strongly more important and absolutely more important, respectively. Even scales of 2, 4, 6 and 8 are used to compromise slight difference between two classifications [20]. Hierarchy structure of risk criteria The final goal of the decision making is the determination of important random variables, which are selected for the risk assessment of the target PSC box girder bridge constructed by a MSS method. Three criteria, used in the XY-comparison matrix method as utilizing construction loads, resistance at each construction phase, and stress at each construction phase are used as the criteria. Table 2 shows the three criteria, which are evaluated by the experts in Korea Rail Network Authority. The AHP evaluation hierarchy framework established by those criteria is shown in Figure 1. Criteria Description

Construction loads · Relationship and risk severity between random variables and construction loads

Resistance at each construction phase

· Relationship and risk severity between random variables and resistance at construction phase

Stress at each construction phase

· Relationship and risk severity between random variables and stress at construction phase

Table 2 The best decision making criteria, determined based on expert’s opinion

Hierarchy

Figure 1 Problem hierarchy in order to determine the contributing random variables Weighting value of each criterion Employing criteria, the design random variables are evaluated for each criterion, as illustrated in the Figure 1, which is a problem hierarchy diagram. On the basis of professional knowledge from the experts, pair comparison and matrix comparison of criterion items at each level in the hierarchy framework are carried out. Also, check for consistency of the eigenvector derived from the comparison matrix and identify the weighting of each criterion item. Since the priority of each element is developed systematically and objectively, they are reliable to provide problem solutions for multi-factors decision-making situations.

Criteria A B C Eigen vector A (Load) 1.00 0.50 0.50 0.3181

B (Resistance) 2.00 1.00 2.00 0.8021 C (Stress) 2.00 0.50 1.00 0.5053

Table 3 AHP-comparison matrix of each criterion As observed in Table 3, since most of input design variables are related with resistance, the output of structural analysis program, such as stress or deflections, are selected for composing limit state functions, which would determine the risk of the considered system. In case the pairwise comparisons are completely consistent, the matrix has rank 1 and the maximum of rank as n. In that case, weights can be obtained by normalizing any of the rows or columns of the matrix. The procedure described above is repeated for all subsystems in the hierarchy. In order to synthesize the various priority vectors, these vectors are weighted with the global priority of the parent criteria and synthesized. This process starts at the top of the hierarchy. As a result, the overall relative priority to be given to the lowest level elements is obtained. These overall, relative priorities indicate the degree to which the alternatives contribute to the focus. These priorities represent a synthesis of the local priorities, and reflect an evaluation process that permits to integrate the perspectives of

the various stakeholders involved. The weights can be determined by the eigenvector method.

Criteria A B C D E F G

A Prestressing force 1.00 0.20 0.50 1.00 0.33 0.33 0.50

B Tendon diameter 5.00 1.00 4.00 5.00 3.00 3.00 4.00

C

Distance from top of the section to neutral

axis

2.00 0.25 1.00 2.00 0.50 0.50 1.00

D

Compressive

strength of concrete

1.00 0.20 0.50 1.00 0.33 0.33 0.50

E Creep coefficient 3.00 0.33 2.00 3.00 1.00 1.00 2.00

F Shrinkage

of concrete

3.00 0.33 2.00 3.00 1.00 1.00 2.00

G Relative humidity 2.00 0.25 1.00 2.00 0.50 0.50 1.00

Table 4 AHP-comparison matrix of each criterion An effective way to obtain group judgments for evaluating a complex problem is using a questionnaire to collect different viewpoints from a number of individuals. The statistics of the group response from the questionnaire may reflect the consensus of opinion and may be used as the basis of evaluation. The questionnaires were returned answered and the result of pair comparison matrices and eigenvectors for setting the priorities among the evaluation criteria in the hierarchy is shown in following Table 5. The weighting factors for each attributes and risk factors are integrated in Table 6.

Risk Factors (Random Variables)

Criteria A (Load) B (Resistance) C (Stress) weighting

factor

Prestressing force

A 1.00 1.00 0.33 0.2000 B 1.00 1.00 0.33 0.2000 C 3.00 3.00 1.00 0.6000

Tendon diameter

A 1.00 0.25 0.33 0.1226 B 4.00 1.00 2.00 0.5571 C 3.00 0.50 1.00 0.3202

Distance from top of

the section to neutral axis

A 1.00 1.00 2.00 0.4000 B 1.00 1.00 2.00 0.4000 C 0.50 0.50 1.00 0.2000

Compressive strength of concrete

A 1.00 0.33 0.33 0.217 B 3.00 1.00 2.00 0.8262 C 3.00 0.5 1.00 0.5199

Creep coefficient

A 1.00 0.33 0.33 0.2279 B 3 1.00 1.00 0.6885 C 3 1.00 1.00 0.6885

Shrinkage of A 1.00 0.25 0.2 0.1145

concrete B 4.00 1.00 0.5 0.3777 C 3.00 5 1.00 0.9188

Relative humidity

A 1.00 0.25 0.33 0.1929 B 4.00 1.00 0.5 0.5588 C 3.00 2 1.00 0.8066

Table 5 Weighting factors of each random variable Risk factors (random

variables) Load Resistance Stress

Prestressing force 0.2 0.2 0.6 Tendon diameter 0.1226 0.5571 0.3202

Distance from top of the section to neutral

axis 0.4 0.4 0.2

Compressive strength 0.217 0.8262 0.5199

Creep coefficient 0.2279 0.6885 0.6885

Shrinkage coefficient 0.1145 0.3777 0.9188

Relative humidity 0.1929 0.5588 0.8066 Table 6 Weighting factors for each attributes The evaluation matrix is determined by the problem hierarchy and criteria weights as the weights of each attributes are multiplied by the weights of risk factors, shown in Table 7.

XY compared matrix value Weighting factor Determined

weighting factor

Random consistency Index (RI)

0.0985 0.6809 9 6.9142 0.1109 0.8463 11 7.6297 0.0894 0.6002 8 6.715 0.1609 1.2003 16 7.4603 0.1562 1.1915 16 7.6272 0.1695 1.2988 17 7.6618 0.2146 1.7504 23 8.1564

0 0 0 0 0 0 1 7.5683 100 8.1564

Table 7 The determined weighting factors Consistency check A measure of consistency of the given pairwise comparison is needed. The consistency is defined by the relation between the entries of A: aij · ajk = aik. The “consistency index” (CI) is

CI = (Rankmax − n)/(n − 1) (1) The final consistency ratio (CR), on the basis of which one can conclude whether the evaluations are sufficiently consistent, is calculated as the ratio of the CI and the random consistency index (RI), final column of Table 7. The number 0.1 is the

accepted upper limit for CR. If the final consistency ratio exceeds the number, the evaluation procedure has to be repeated to improve consistency. The measurement of consistency can be used to evaluate the consistency of decision makers as well as the consistency of all the hierarchy.

CR = CI/RI. (2)

Evaluation item

Prestressing force

Tendon diameter

Distance from top of the section to neutral

axis

Compressive strength

Creep coefficient

Shrinkage coefficient

Relative humidity

1)C.R. 0.5272 0.6476 0.5491 0.9944 0.9726 0.8036 0.9171 1)C.R. < 0.1 : verifying the consistency of the analyses and evaluations Table 8 Consistencies for the evaluation Consequently the analyzed results reveals the order of importance as: Compressive strength, Relative humidity, Shrinkage coefficient, Tendon diameter, Distance from top of the section to neutral axis, Prestressing force, Creep coefficient, as can be observed in Table 8. As regarding relatively short manufacturing time for the segment of PSC box girders, we have selected the shrinkage coefficient as a random variable more important than creep coefficient. Accordingly, compressive strength, relative humidity, and shrinkage coefficient are selected for the important random variables in the reliability analysis of PSC box girder bridges during the construction phases.

Reliability analysis by an adaptive weighted response surface method

An improved response surface method Due to the merits discussed in the previous section, the basic response surface method has been widely applied to the reliability analysis owing to the merits discussed in the previous section. The approximation of structural responses, however, would show relatively large errors depending on the form of nonlinearity of the limit state functions [21]. Bucher [13] proposed an adaptive response surface method in order to overcome the drawback of RSM by narrowing the distance between the design points and the original limit state surface (g()=0) using a linear interpolation as:

M D( )X (X )

( ) ( )D

g XX Xg X g X

= + −−

(3)

where XM is the new center point selected on a straight line from to XD, and XD indicate the mean value of the random variables and the design point obtained from the first (or previous) stage, respectively. Another meritorious approach was performed by Kim and Na [22]. Based on the gradient projection method, they significantly increased the accuracy using a linear response surface function instead of a conventional quadratic response surface function. However, they used Latin Hypercube Sampling (LHS), a modified Monte-Carlo Simulation, which may still need over hundred times of simulations. Meanwhile Irfan et al. [21] proposed a weighted regression method in which the response surface function, formulated by assigning higher weights to the points closer to the limit state. To assign higher weights to the points closer to the limit state, he adopted a n x n diagonal matrix of weights (Eq. (16)), and the best design among the responses from the performance function corresponding to the design matrix is selected based on closeness of the limit state function to the value of zero:

f min( ( ) )best ig x=r

(4) where f best

r is the smallest value among the calculated values of the limit state function, g(x)i; i

means ith random variable Thus, the weight for each experiment is given by the following formula:

( ) fw expf

bestii

best

g x⎛ ⎞−= −⎜ ⎟⎜ ⎟

⎝ ⎠

r

r (5)

After the determination of weights, the adaptive design points and the new axial points can be obtained by an adaptive RSM (Bucher’s Eq.(3)) or Rackwitz-Fiessler method. Although the weighted regression method results in great improvement of RSM, first of all, Irfan et al. divided the design space to 4 quarters using sign conventions, which produce additional stages. Second of all, Eq.(3) may diverge depending on the linearity of limit states due to the limitation of linear interpolation. The improved techniques over previous studies are compared in Table 9.

Improved techniques for

RSM

This study RSMM [23] Kim and Na [22]

Irfan et al. [21]

1) Adaptive method

Rackwitz-Fiessler method

Adding experimental point

Rackwitz-Fiessler method

Bucher’s (Eq.(3))

2) Weight method

Exponential form (Eq. (5))

N.A. N.A. Exponential form (Eq. (5))

3) Linearity of response surface function

Linear Quadratic, and inserting cross product terms

Linear Quadratic

Table 9 Comparison of the improvement in the techniques for determining the coefficients of response surface

Comparing the adaptive methods, to improve the convergence and accuracy in nonlinear limit state functions, inserting weight matrix with adaptive iteration by the Rackwitz-Fiessler method, the linear adaptive weighted response surface (LAWRS) function is determined. The best combination of the improved techniques compared in six cases of combined methods (both linear and nonlinear response surface functions using basic, adaptive, and adaptive weighted method) was determined as the liner adaptive weighted linear response surface method (LAW-RSM), which is verified in the next examples. 2.5 Validation of the LAW-RSM in a highly nonlinear limit state function This example is the first case of Example 2 from the cited reference [21] in which the limit state is defined to investigate the dependency of convergence among the discussed response surface techniques. The highly nonlinear limit state is defined as:

3 2 31 1 2 2( ) 18g x x x x⋅ = + + − (6)

The mean and standard deviation of the considered random variables are listed in Table 10 Parameters of random variables in Example2

Index Name of variable Mean *C.O.V. Type of distribution

1 X1 10 0.5 Normal 2 X2 10 0.505 Normal

Table 10 The statistical values of the input random variables

Analysis Method Reliability Index

Probability of

Failure

Error (%)

Comments

MCS 2.533 0.006 0.000 The number of simulation=1,000,000

Linear RSM

1. Basic 0.923 0.178 6356.1 2. Adaptive 2.79 0.003 -10.1 Rackwitz-Fiessler

3. Adaptive Weighted

2.722 0.003 -7.5 After 8th iteration of Rackwitz-Fiessler with Exponential weighting

Quadratic RSM

4. Basic 1.086 0.139 57.1 5. Adaptive 1.878 0.03 25.9 Rackwitz-Fiessler 6. Adaptive

Weighted 1.988 0.023 21.5 After 15th iteration of

Rackwitz-Fiessler with Exponential weighting

Table 11 Comparison of the converged reliability among various response surface functions The LAW-RSM showed the minimum difference of 7.5% with the result of MCS, among the compared results by five other RSMs. From the responses of the nonlinear problem, the fitted functions behaved nonlinearly around the design point (Figure 2). First of all, the linear and nonlinear adaptive weighted RSMs show better convergence than four other methods. Compared with the results of MCS, the reliability index by the linear basic RSM showed bigger differences than by quadratic RSMs. However, the linear ARSM converged rapidly with the update of the design matrix by adaptation with or without weighing method. Adaptive calculation reduces the differences from 6356% to 10.1%. And the weighting method reduces the difference 2.6% more. It might be mainly due to the unsymmetrical form of the limit state function, which is a cubic form, thus a quadratic form of RSM could not access to the failure point less than 20% of difference, even after 15th iterative calculations in the example.

Figure 2. Comparison of the converged reliability among various response surface functions (g0: the limit state function in black line; g1: Basic Linear RSM in blue line; g2: Linear Adaptive RSM in brown line; g3: Linear Adaptive Weighted RSM in violet line; g4: Basic Quadratic RSM in yellow line; g5: Quadratic Adaptive RSM in red line; g6: Quadratic Adaptive Weighted RSM in green line)

Reliability analysis for the construction phases of PSC box girder bridge

Description of the target bridge system A three span continuous PSC box girder railway bridge is modeled for performing reliability analysis of the bridge in terms of the linear weighted response surface method. The PSC high-speed railway bridge has 14m wide with ballasted double tracks having the eccentricity of 2.5 m, and 40m span length each. The cross-section details of the bridge are shown in figure 3, while

material properties of the bridge are given in table 12. The analytical modeling employing 12400 elements with 12366 nodes are shown in figure 4.

Figure 3 Cross-section details of the three span continuous PSC box-girder railway bridge

Remark Design value

Concrete Young’s modulus(kN/m2) 29,430,000

Poisson’s ratio 0.15 Specific weight(kN /m3) 24.53

Permanent weight of track structure (kN /m) 65.67

Table 12 Material properties of the bridge

(a) 3-D view (b) Mesh refinement for deck Figure 4 Finite element analysis model for the PSC box-girder bridge Finite element modeling for a construction phases The construction phases are modeled in order to evaluate the tensile stresses at top and bottom of PSC box girder and prestressing losses at the following steps, illustrated in Table 13.

stage Step day Tasks at the construction phase

1 - 27 The first span built

1 28 Prestressing tendon #1 and #2

2 30

Moving MSS to the second span Installing external scaffolding and prestressing tendons Assembling re-bars and ducts Installing internal scaffolding

3 40 Casting concrete of the second span

2 - 43 The second span is constructed

4 44 Prestressing tendon #3 to #6

5 50 Prestressing remaining tendons #6 to #12

Table 13 Comparison of the converged reliability among various response surface functions The construction phases, from 27th to 50th day construction phases are selected as observation of the stresses and prestressing losses in order to assess the probability of failure by simulating the considered limit state functions. The limit state functions are composed by utilizing the proposed LAW-RSM. To compose response surface, the selected input random design variables, the observed stresses, and prestressing loss values. The process of evaluations are described in next sections. Risk assessment for the construction stages of the PSC box girder bridge Random variables are assumed to be uncorrelated. If the limit state function is less than 0, it means the failure of the considered system or the violation of rupture stress in code specified value or an expected prestressing loss [24]. Table 14 below shows the statistical properties of the selected random variables.

Random Variables Index Mean value C.O.V. Reference Compressive strength X1 39.24 MPa 0.066 [25]

Relative humidity X2 70% 0.269 [25] Shrinkage coefficient X3 2 0.542 [26]

Stress in lower flange at midspan X4 3.90 0.100 Assumed Stress in upper slab at pier X4 3.90 0.100 Assumed Prestressing loss at transfer X4 6.77 0.069 [24]

Prestressing loss after all X4 19.53 0.069 [27] Table 14 Random variables and statistical values The initial design values of random variables for composing response function use initial design matrix as the average of random variables and three axial points. The three axial points had the distances of ± (standard deviation) between the center and axial points. The selected random variables are determined by AHP method, described earlier, as the demand terms in limit state function. The supplying terms in limit state function are tensile rupture stress of concrete on top and the bottom of deck plate. and target loss of prestressing force of 6.77% [24] as an immediate loss at transfer and 19.53% as the all losses [27]. For the three input variables, 7 cases of analyses provided responses (output of FEA) of the bridge as stresses and prestressing losses. Among the construction phases from 28th to 50th days, the input and output values of the first and the last day are presented in Table 15 and Table 16.

Case

Input random variables Calculated responses

X1, Compre

ssive strength

X2, Relative humidity

X3, Shrinkage coefficient

Stress in lower

flange at midspan

Stress in upper slab

at pier

Prestressing loss (%)

CASE1 4000 70 2 -

1.2480E+02

-1.3000E+0

2 0 0

CASE2 4264 70 2 -1.248E+02 -1.300E+02 0 0 CASE3 3736 70 2 -1.248E+02 -1.300E+02 0 0 CASE4 4000 88.83 2 -1.248E+02 -1.300E+02 0 0

CASE5 4000 51.17 2 -1.248E+02 -1.300E+02 0 0 CASE6 4000 70 3.084 -1.248E+02 -1.300E+02 0 0 CASE7 4000 70 0.916 -1.248E+02 -1.300E+02 0 0

Table 15 Axial points as the input values to LAW-RSM for constructing response surface functions (ton-m)

Case

Random variables Observation points in spans

X1, Compre

ssive strength

X2, Relative humidity

X3, Shrink

age coeffici

ent

Stress in lower

flange at midspan

Stress in upper slab

at pier

Prestressing loss (%)

CASE1 4000 70 2 -

2.2230E+02

-2.1490E+

02 10.45 10.34

CASE2 4264 70 2

-2.223E+02-

2.149E+02

10.23 10.12

CASE3 3736 70 2

-2.221E+02-

2.148E+02

10.7 10.6

CASE4 4000 88.83 2

-2.219E+02-

2.147E+02

10.89 10.79

CASE5 4000 51.17 2

-2.223E+02-

2.149E+02

10.32 10.21

CASE6 4000 70 3.084

-2.212E+02-

2.142E+02

10.8 10.7

CASE7 4000 70 0.916

-2.233E+02-

2.156E+02

10.11 9.99

Table 16 Axial points as the input values to LAW-RSM for constructing response surface functions (ton-m) The limit state functions for the risk assessment The construction stage-dependent limit state functions for the rupture of concrete on top and bottom of concrete box girder may be expressed:

( , ) 3.9 ( )ig i t tσ= − (7) where i is the location of observation (top and bottom of the box girder), t is the curing date during the construction, iσ is the tensile stress at the top and bottom of PSC box girder(MPa), and Modulus of rupture for concrete is determined as [28]

fr=7.5 cf (psi) (8)

= 7.5 39.24 / 6.894 /1000 0.566 3.900MPa ksi MPa= =

Using the variation of random variables in Table 16, the coefficients of response surfaces are determined, after updated iteratively and converged by the LAW-RSM, to compose the limit state functions during the construction stages as follows: Risk of the tensile crack in lower flange at midspan:

161 2 3( ,28) 3.9 (1.224 0 1.603 10 0 )g low x x x−= − + ⋅ − ⋅ + ⋅ (11)

4 4 61 2 3( ,42) 3.9 (1.205 1.931 10 1.328 10 7.756 10 )g low x x x− − −= − + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅

(12) 4 4 5

1 2 3( ,50) 3.9 (2.172 3.861 10 1.062 10 1.086 10 )g low x x x− − −= − + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ (13) The above equations are fitted functions for the limit state of rupture of concrete in upper and lower flange of concrete box girder during the construction stages by employing LAW-RSM at the center of main span (Eq. (7)). Risk of the tensile crack in lower flange on pier:

15 16 181 2 3( ,28) 3.9 (1.275 4.237 10 1.603 10 1.695 10 )g up x x x− − −= − − ⋅ ⋅ − ⋅ + ⋅ ⋅

(14) 4 5 6

1 2 3( ,42) 3.9 (1.248 1.931 10 5.311 10 5.688 10 )g up x x x− − −= − + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ (15)

4 5 61 2 3( ,50) 3.9 (2.104 1.931 10 5.311 10 7.239 10 )g up x x x− − −= − + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅

(16) When applying LAW-RSM by adaptive iteratively with weighting matrices, the convergence is reached if the difference of reliability indices is less than 0.1%. After constructing limit state functions, the reliability indices and the probability of failure are evaluated by first order second moment method (FOSM) of Rackwitz-Fiessler method [29]. Successively, the risk of tensile crack on the box girders are evaluated as increased probability of failure from 0 to 4.7•10-10 at the end of the construction step, Prestressing remaining tendons #6 to #12 (Table 13). As observed in Figure 5 and Table 17, the probability of tensile cracks during the considered construction phases is very low, which could be shown in reliability indices in Figure 5 as well.

5

6

7

8

9

10

11

12

30 35 40 45 50

beta_lowbeta_up

DAY

-1 10-9

0

1 10-9

2 10-9

3 10-9

4 10-9

30 35 40 45 50

Pf_lowPf_up

DAY (a) reliability indices (b) probabilities of failure

Figure 5 The converged reliability indices and probabilities of failure during the construction stage

DAY lowβ upβ _F lowP _F upP

28 11.57 11.16 0 0

42 11.74 11.34 0 0 50 5.78 6.112 3.713E-09 4.679E-10

Table 17 The converged reliability indices and probabilities of failure during the construction stage Risk assessment for the prestressing loss The limit state functions for the loss of prestressing force are prepared for the instantaneous and long term losses as:

( , ) 6.77 ( )ig i t P t= − (17)

( , ) 19.53 ( )ig i t P t= − (18) where i is the number of tendon, t is the curing date during the construction, the target 6.77% loss of prestressing force [24] is an immediate loss at transfer, 19.53% is the targeted all losses [27], and iP is the loss of prestressing force in a strand at each construction stage (%). Among 12 tendons, the first and second one may show the largest loss of prestressing forces. Hence, the two tendons are considered for the evaluation of risk for the loss. The converged fitted functions for the first and second prestressing tendon after transfer and built of 2nd span are calculated as: Composed limit state function with supplying term, loss of prestressing at transfer as 6.77%: For the first tendon:

2 3 41 2 3(1,28) 6.77 (5 2.12 10 6.64 10 2.59 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (19)

2 3 41 2 3(1,29) 6.77 (5 2.22 10 6.67 10 2.78 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (20)

For the second tendon: 2 3 4

1 2 3(2,28) 6.77 (5 2.32 10 6.64 10 2.64 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (21) 2 3 4

1 2 3(2,29) 6.77 (6 2.51 10 6.70 10 2.84 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (22) Composed limit state function with supplying term of after all losses as 19.53%: For the first tendon:

2 3 41 2 3(1,29) 19.53 (5 2.22 10 6.67 10 2.78 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (24)

2 2 41 2 3(1,50) 19.53 (13 9.07 10 1.51 10 3.57 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ (25)

For the second tendon: 2 3 4

1 2 3(2,29) 19.53 (6 2.51 10 6.70 10 2.84 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ + ⋅ ⋅ (26) 2 2 4

1 2 3(2,50) 19.53 (13 9.27 10 1.54 10 3.67 10 )g x x x− − −= − − ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ (27) The constructed limit state functions are evaluated by FOSM (Rackwitz-Fiessler method) for the reliability indices and the probability of failure. As observed in Figure 6, the probability of exceeding 6.77% as loss of prestressing force is increased from 0 to 4.93•10-6 at the 2nd date of prestressing in tendon number 1. The probability after all loss show the lower value as 2.612•10-9, after 50 days from prestressing.

0

2

4

6

8

10

12

14

30 35 40 45 50

beta_P1_all

beta_P2_all

beta_P1_trans

beta_P2_trans

DAY

-2 10-6

0

2 10-6

4 10-6

6 10-6

8 10-6

1 10-5

30 35 40 45 50

Pf_P1_allPf_P2_allPf_P1_transPf_P2_trans

DAY Figure 6 The converged reliability indices and probabilities of failure during the construction stage after instantaneous and long term loss of prestressing forces Conclusion Based on the results of analyses, the following conclusions are drawn: 1) The important input random variables were determined based on a decision making process by employing AHP method. Which provides more objective and quantified alternatives than the conventional subjective decision process. By ranking for important variables, compressive strength of concrete, relative humidity, and shrinkage coefficient are selected for the evaluation of risks during construction processes of the target bridge system. 2) An improved linear adaptive weighted response surface method (LAW-RSM) has been developed. The LAW-RSM shows the best converged result, regardless of the linearity of limit state functions, than the other five combined RSM methods. 3) Regarding the risk of tensile crack on the concrete box girders during movable scaffolding method constructions, the minimum reliability index was 5.781, which is rather a reliable state during this construction method. 4) The risk of prestressing loss during the construction process is compared with a mean value of current design practice. The probabilities of failure for the evaluated result show less than 4.93•10-6, which means higher than 4.3 of reliability index at transfer state, and higher than 5.8 of reliability index after all losses considered. In all, based on the present risk assessment, the target PSC box girder bridge have enough margin during the construction by MSS. The employed AHP and LAW-RSM have provided more objective and cost-effective alternative by modeling statistical variations in materials, structures, and construction methods.

Acknowledgements

This work was supported by the research grant from MOCT (Ministry of Construction and Technology) of Korean Government (Grant number: 06HIGH-TECH FUSION-E01). The authors hereby express their sincere appreciation.

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