Probabilistic risk assessment for the construction phases of a bridge construction based on finite element analysis

Finite Elements in Analysis and Design 44 (2008) 383 – 400 www.elsevier.com/locate/finel

Probabilistic risk assessment for the construction phases of a bridge construction based on finite element analysis Taejun Cho a , Tae Soo Kim b,∗ a KR Technical Research Institute, Korea Rail Network Authority, Daejeon 301-722, Republic of Korea b Sustainable Building Research Center, Hanyang University, 1271, Sa-1-dong, Sangnok-Gu, Ansan, GyeongGi-do 426-791, Republic of Korea

Received 20 August 2007; received in revised form 2 December 2007; accepted 3 December 2007 Available online 31 January 2008

Abstract To develop a design, engineering, and construction management information sharing system that allow the project participants to effectively share the information throughout the construction life cycle with the support of 3D, design and building information, a “virtual construction” research project started in Korea. As a part of virtual simulation of construction processes, this paper deals with the quantitative risk assessment for the construction phases of the suspension bridge. The main objectives of the study are to evaluate the risks in a suspension bridge by considering an ultimate limit state for the fracture of main cable wires and to evaluate the risks for a limit state for the erection control during construction stages. While many researches have been evaluated the safety of constructed bridges, the uncertainties of construction phases have been ignored. Therefore, the statistical variation of input random variables is considered based on the quantitative results of finite element analysis for the construction phases of an example suspension bridge. The analyzed results have been compared with the conventional safety indices and allowable error for the control of deformations during construction. 䉷 2007 Elsevier B.V. All rights reserved. Keywords: Risk assessment; Virtual construction; Construction phases; Improved response surface method; Finite element analysis

1. Virtual construction 1.1. Introduction To maximize the productivity and quality control of construction, the current goal of virtual construction (VC, hereafter) or computer integrated construction (CIC, hereafter) include sharing and re-using spectral and stochastic information, as in a form of database for the whole information relating planning, design, construction, and maintenance of infra-structures, which have been accumulated but ineffectively or seldom used. For the purpose, a VC research team started to develop an integrated computer system, using a name of the VirtuAlmighty system (VA, hereafter) in Korea. The VA system consists of three parts for a 3D-design system, for a construction process simulation, and for an integrated decision making system. Civil and Architect team have three parts respectively, and ∗ Corresponding author. Tel.: +82 31 400 4690; fax: +82 31 406 7118.

E-mail address: [email protected] (T.S. Kim). 0168-874X/$ - see front matter 䉷 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2007.12.004

the civil engineering parts are shown in schematic diagram as Fig. 1. To verify the developed individual simulation programs until 2011, four pilot test structures are modeled each year. The first one (a) in Fig. 2 is the example bridge for the probabilistic risk assessment for VA (VA-Risk, hereafter) in this study. By the VA-Risk data of user-selected members, location, and specific critical construction stage after the evaluation of risk assessment, the results, probability of failure, will be delivered to the visualization program, VA-ciconsimulator, visual simulation of construction processes (2nd part in Fig. 1). The integrated database is usually presented in n-dimensional CAD. In VA system, VA-Risk system could be called as 6D CAD, after added to the 5D CAD. 3D and 4D CAD system is now widely provided by most major CAD companies. In general, 4D CAD is composed of modeling, developing objects, simulation construction sequences and check interfaces, which have been realized by Bentley’s Schedule Simulator, Four Dviz’s Visual Project Schedule, Virtual STEP’s 4D-planner, or Intergraph’s Schedule Review. Integrating information for construction process has been shown by EuroSTEP’s

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1ST PART

Virtual Design System

2ND PART

VA - Bridge VA - Road VA - Rail VA - Flibrary

VR construction simulation

VA Ciconsimulator

3RD PART

Integrated decision making system Construction Project Life-Cyde Management System

Virtual Automatic Estimating

VA - Estimator

Risk Assessment System for the Construction Phases of Structures

VA - Risk

3D VC data compatibility and standardization

3D Design Guidelines

VA - CPLM VA - Decsupporter VA - PMIS VA - Library VC - Academy

VA - SDAI (K-IFC)

VA - CGuide

Fig. 1. The organization of VA and the developing modules (Civil engineering fields).

Fig. 2. Test-bed structures, which will be modeled via VA within next four years. Table 1 Comparison among DB systems for users and data sharing The previous DB or standardization programs

Report to owner from company

Report to company from owner

Sharing information

Users

CALS/CITIS PMIS Computer integrated construction/ virtual construction

Yes No Yes

No No Yes

Expanded Yes Yes

Government from 1998 Public from 1997 Government and public from 1988

4D-Linker. And Microsoft Project or Primavera’s P5 can be an example of project planning. It has been expanded to 5D form by Vico software by adding cost estimator to 4D CAD. The main difference compared with the previous standardization or database sharing systems could be explained by comparing the owner and the user of developed system. In Table 1, CALS, meaning a continuous acquisition and lifecycle support, as an operational concept started by governments, integrates all the processes of design, manufacturing, procurement, and managements of whole goods of the manufacturing industry. All the information in CALS is exchanged in an automated, standardized, and digitalized form, thus it is a paperless system. Embodied in a contractor integrated technical information service (CITIS, hereafter), the CALS is now

expanded and connected with the non-government information systems. Project management information system (PMIS, hereafter) is very similar with the “virtual construction”, involving construction process control, cost management, quality control, contract and procurement management, design and document management, and safety or environmental management. The problems in the previous developed standardized systems for sharing construction information were reported several times [1,2], which are: (1) incomplete integration of individually developed modules, (2) lack of interactive exchanging information between organizations, and

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(3) incomplete standardization, which may not be completely accomplished, due to its character of updated needs of users, discussed more detail in Section 2.1. 1.2. Objectives of VA-Risk for the virtual construction project Based on the discussed pros and cons of previous developed standardized information sharing programs, the final goals of VA are (1) increased communication among the developed individual modules, which include the planning, conceptual design, structural analysis, detail design, construction simulations using VR, and automatic estimating; (2) embodiment of the integrated virtual construction system in n-dimensional CAD system, where 3D CAD, added by time, cost, and risk assessment, makes 6D CAD system, which could be updated to n-dimensional form; (3) define standardized rules and expressions considering interfacial needs among developed modules. The VA-Risk also seeks the common goals of VA, with special interests on the risks during construction process. Based on the recent survey for the causes of accidents in Korean railways [3], it is found that 3% of them are caused at the planning stage, and 78% of them are caused by the risk events during construction. The accidents throughout the service duration account for only 19% of the 80 accidents on railway structures. It is also found that 67.5% of the accidents are caused by human errors, which include the violation of safety regulations (35%), personal carelessness (22.5%), error in communications (10%), etc. Consequently, over 81% of the accidents related to the uncertainties in human error, planning, design, and material and loads during the construction have occurred before the completion of construction. Therefore, more research is necessary to find the causes and solutions for the risks during construction phases. The objective of VA-Risk for the construction phases of structures considering both the results of recent survey for the accidents in Korea and the interfaces among developed simulation modules in virtual constructions are as follows: (1) To develop quantitative risk assessment method, interactively working with structural analysis programs in VA, e.g., finite element (FE) analysis. (2) To develop probabilistic and quantitative risk assessment for the life cycle of structures, focusing on construction phases. (3) To apply the developed VA-Risk model in a pilot test structure, using the developed improved response surface method (RSM) that converges in less numbers of stages in adaptive calculations, regardless of evaluating for linear or nonlinear limit state functions. 2. Risk assessment in the virtual construction 2.1. Interface between virtual construction and the VA-Risk A comparison of VA with the previously developed programs, focused on standardization, is presented in Table 2. CALS includes categorizing steps of construction attaching naming

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codes, developing business procedures for public constructions, defining products of each construction steps, developing standard system related with International Organization for Standardization (ISO) for the classified quality management, CITIS for exchanging information, IFC for exchanging standardized information for wide applications in construction including 3D CAD design, where IFC is Industry Foundation Class. The standardization of data for construction information system has been studied by many research teams world widely. Among them, ISO/STEP and IAI/IFC are the most promising ones, which are compared with VC in Table 2. In Table 2, STEP is a STandard for the Exchange of Product model data, which models product’s life cycle with process of manufacturing, as a part of ISO now applied widely in architect, plant, and other structural engineering fields, where IAI means International Alliance of Interoperability, which started 1994 and approved by ISO since 2002. Including IFC, IAI is the worldwide standard for sharing information about industries of AEC (Architecture/Engineering/Construction). Over 70% of computer software of AEC in the world selects to use IAI/IFC. As compared in Table 2, VA will provide standardized protocol among the distributed developed systems, such as estimator or automated-structural optimum design system. Fig. 3 depicts a conceptual image of the standardized interfacial protocol among VA modules. The developing protocol is already applied for the portal internet websites, such as yahoo.com or amazon.com successfully, by providing standard input and output format of data, which are for the 3rd party developers. The advantage of the protocol radiates whenever new system is added to the old integrated system. If error messages are appropriately provided, because the protocol exactly tells the location and program violates the standard format, the error can be handled by developers in the shortest time and efforts without updating old modules. 2.2. VA-Risk Risk assessment is a qualitative, quantitative, and financial concept expressed as the probability of risk that can happen in the future. In general, the risk is determined as the probability of risk multiplied by the financial loss. If compared with conventional safety concept, safety means a qualitative concept used in wide area such as safety about foods, safety of risk for indebt, which need to be managed within the allowed level. The modeling of uncertainties in material, load, and construction, considering construction phases by the prediction of probability of failure needs probabilistic distribution of the variables, for the final goal of decision making tool for selecting the best choice among construction methods. The best choice is determined based on the cost and safety, predicted by the VA-Risk analysis system, which should be determined in shorter time, could be applied to any system interfacing, interactively working through internet by the server system and client computer. To assess risks during construction, commercial programs are already used in market. But it is limited to construction managements with a few analysis techniques such as Monte

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Table 2 Comparison among DB systems for standardization and data sharing The previous standardization programs

Standardization

Applications

Data exchange

Network

CALS/CITIS ISO 10303 (STEP) IAI/IFC VA/virtual construction

2D drawing and documents Manufacturing and process 3D information 3D information, supporting 2D drawing

Applicable if transferred Applicable if expanded Applicable if expanded Applicable as providing standardized protocol for the integrated system

STEP supporting XML Applicable if expanded STEP supporting XML STEP supporting XML

Unlimited Unlimited Unlimited Unlimited

Fig. 3. Interactive data communication among the developed systems using a proposed standardized interface service program. Table 3 Comparison of the improvement in the techniques for determining the coefficients of response surface Improved techniques for RSM

This study

RSMM [32]

Casciati and Faravelli [30]

Irfan et al. [22]

(1) Adaptive method (2) Weight method (3) Linearity of response surface function

Rackwitz–Fiessler method Exponential form (Eq. (3)) Linear

Adding experimental point N.A. Quadratic, and inserting cross product terms

Rackwitz–Fiessler method N.A. Linear

Bucher’s (Eq. (1)) Exponential form (Eq. (3)) Quadratic

Carlo simulation (MCS) or basic RSM, which neglects co-operations with CAD, structural analysis, estimating, and construction. Furthermore, for simulating the spatial and time-dependent variability of uncertainties, MCS is usually employed to any system, regardless of correlation among components. However, 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 [4], applied to MCS [5,6], Latin hyper cube method [7,8], and Markov chain modeling [9,10] have been developed. One of another approach accounting for uncertainties is stochastic finite element method (SFEM, hereafter), the modified FEM for spectral and stochastic random fields. SFEM could solve the limitation of MCS by a perturbation technique [11,12] or by a weighted integral method [13,14] 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 has 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. 2.3. VA-Risk for the construction phases of a long span bridge The measure of safety for the construction of a suspension bridge will be a function of the variability in loads and resistance and the estimate of the probability of exceeding design criteria during the design life as for more uniform level of safety throughout the system. Probabilistic design concept is widely applied to the design of bridges to provide the same measure of reliability for all current and future materials and construction methods, which can be repaired or replaced. While geometric nonlinear analyses [15,16] or a vehicle– bridge interactive nonlinear vibration analysis [17] in the reliability evaluations for the global behavior of a suspension

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

bridge have been studied, allowable stress is still used in its design and construction [18]. The main cable wire shows a nonlinear constitutive behavior during its large deformation under severe load conditions such as earthquake load or wind load. Therefore, the design of a suspension bridge needs to check the ultimate limit state of the components during and after the construction. Not only the safety of a structural system should be maximized at the lowest price, but the risk management of civil structures should be established based on the consideration of probabilistic variations in resistance and loads as studied by many researchers [19–21]. Erection control of the structural component, which is built through many construction stages including series connections, especially, of free cantilever segments during construction of prestressed concrete bridges, is crucial. Regarding a displacement loading, the fault in erection stage has the same effect to the structural system as the fault in fabrication stage. The construction of the target structure of this study, a cable suspension bridge, is composed of a series of connection among components, which can be under severe vertical and lateral loads. Therefore, a precise prediction and control for its erection phase is mandatory for this type of structures. Table 4 Random variables and statistical values (N-mm) Random variables

Notation for the random variables

Mean value

C.O.V.

Distribution

As D fy Mu

x1 x2 x3 x4

87.972–351.9 300 400 7.30E06

0.015 0.058 0.015 0.12

Normal Normal Normal Normal

C.O.V.: coefficient of variation. Table 5 Parameters of random variables in Example 2 Index

Name of variable

Mean

C.O.V.

Type of distribution

1 2

X1 X2

10 10

0.5 0.505

Normal Normal

C.O.V.: coefficient of variation.

387

2.4. An improved RSM The basic RSM has been widely applied to the reliability analysis for the various types of structures owing to the merits [19,22–30]. The approximation of structural responses, however, would show relatively large errors depending on the form of nonlinearity of the limit state functions [31]. Bucher [24] proposed an adaptive RSM 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: XM = X + (XD − X)

g(X) g(X) − g(XD )

,

(1)

where XM is the new center point selected on a straight line from X to XD , and X and XD indicate the mean value of the random variables and the design point obtained from the first (or previous) stage, respectively. A refined approach was performed by Casciati and Faravelli [30]. The study employed a linear adaptive response surface function rather than a conventional quadratic response surface function, and thus accuracy was significantly increased. Meanwhile, Irfan et al. [22] proposed a weighted regression method in which the response surface function was formulated by assigning higher weights to the variable that was closer to the limit state. To assign higher weights to the variable, Irfan et al. adopted n × n diagonal matrix of weights, and the best design amongst the responses from the performance function corresponding to the design matrix was selected when the limit state function approaches to zero. Thus, the best response can be given as below, − → f best = min(g(x)i ), (2) − → where f best is the determined best value amongst the axis points, and g(x)i is the fitted response on the response surface composed of center and axis points at the iteration step. The component of weight matrix in iteration can be expressed like the following equation:  − →  g(x)i − f best wi = exp − , (3) − → f best

Table 6 Comparison of the converged reliability among various response surface functions Analysis method

Reliability index

Probability of failure

MCS

2.533

0.006

1. Basic 2. Adaptive 3. Adaptive weighted

0.923 2.79 2.722

0.178 0.003 0.003

6356.1 −10.1 −7.5

4. Basic 5. Adaptive 6. Adaptive weighted

1.086 1.878 1.988

0.139 0.03 0.023

57.1 25.9 21.5

Linear RSM

Quadratic RSM

Error (%) 0.000

Comments The number of simulation = 1, 000, 000 Rackwitz–Fiessler After 8th iteration of Rackwitz–Fiessler with exponential weighting Rackwitz–Fiessler After 15th iteration of Rackwitz–Fiessler with exponential weighting

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Table 7 The limit state function composed of the coefficients of response surface function Analysis method

Limit state functions

Reliability index, 

1. 2. 3. 4.

g(·) = (−8946.36 + 722.99x1 + 619.03x2 ) − 18 g(·) = (−21.43 + 225.00x1 + 225.01x2 ) − 18 g(·) = (−129.86 + 225.86x1 + 225.56x2 ) − 18 g(·) = (−1375.04 − 75.00x1 + 100.00x2 + 39.89x12 − 1.84 × 10−8 x22 ) − 18

0.923 2.790 2.722 1.086

5. Quadratic adaptive

g(·) = (−1545.21 + 183.62x1 + 207.01x2 + 14.39x12 + 9.77x22 ) − 18

1.878

6. Quadratic adaptive weighted

g(·) = (−1350.55 + 192.31x1 + 213.38x2 + 12.64x12 − 8.22x22 ) − 18

1.988

Linear basic Linear adaptive Linear adaptive weighted Quadratic basic

Table 8 The converged coefficients of the responses surface function composing limit state functions of Table 7 through adaptive iterations by linear RSMs Run number Reliability index Probability of failure, Pf

1 0.923 8.220E − 01

2 1.523 9.361E − 01

3 1.925 9.729E − 01

4 2.387 9.915E − 01

Coefficients of response surface function

1 2 3 4

−8.946E + 03 7.230E + 02 6.190E + 02 1.177E − 15

−3.847E + 03 4.388E + 02 4.082E + 02 1.381E − 16

−1.969E + 03 3.229E + 02 3.093E + 02 1.047E − 15

−7.224E + 02 2.466E + 02 2.431E + 02 −3.411E − 16

Best fit (Eq. (2))

Y=

−4.503E + 03

−2.893E + 03

−1.956E + 03

−1.334E + 03

Weight matrix

1 2 3 4

7.342E + 00 1.000E + 00 1.000E + 00 1.665E + 00

7.698E + 00 1.000E + 00 1.000E + 00 1.550E + 00

7.715E + 00 1.000E + 00 1.000E + 00 1.545E + 00

7.598E + 00 1.000E + 00 1.000E + 00 1.581E + 00

Design matrix

x1 x2

6.458 6.868

4.368 4.661

2.978 3.177

2.060 2.149

Normalized design matrix

z1 z2

−0.708 −0.606

−1.126E + 00 −1.048E + 00

−1.404E + 00 −1.345E + 00

−1.588E + 00 −1.550E + 00

5 2.519 9.941E − 01

6 2.612 9.955E − 01

7 2.677 9.963E − 01

8 2.722 9.968E − 01

Run number Reliability index Probability of failure, Pf Coefficients of response surface function

1 2 3 4

−4.712E + 02 2.351E + 02 2.332E + 02 1.440E − 15

−3.103E + 02 2.296E + 02 2.286E + 02 1.689E − 14

−2.028E + 02 2.271E + 02 2.265E + 02 −3.388E − 15

−1.299E + 02 2.259E + 02 2.256E + 02 −1.439E − 15

Best fit (Eq. (2))

Y=

−9.241E + 02

−6.365E + 02

−4.367E + 02

−2.952E + 02

Weight matrix

1 2 3 4

7.467E + 00 1.000E + 00 1.000E + 00 1.623E + 00

7.262E + 00 1.000E + 00 1.000E + 00 1.692E + 00

6.995E + 00 1.000E + 00 1.000E + 00 1.790E + 00

6.619E + 00 1.000E + 00 1.000E + 00 1.945E + 00

Design matrix

x1 x2

1.416 1.439

0.969 0.942

0.652 0.596

0.429 0.355

Normalized design matrix

z1 z2

−1.717E + 00 −1.692E + 00

−1.806E + 00 −1.792E + 00

−1.870E + 00 −1.861E + 00

−1.914E + 00 −1.909E + 00

where wi is the component of weight matrix, need to be multiplied to the center or axis points, for composing an improved response surface of the next iterative step. Table 3 summarizes the improved techniques with reference to previous studies regarding RSM. As shown in Table 4, RSMM [32], Casciati and Faravelli [30], and Irfan et al. [22] approaches showed well-converged analysis results in a limited case, e.g., a linear limit state function. However, for

the application to the nonlinear limit state function, linear and weighted combination technique may lead to the best solution. This is due to the fact that the quadratic RSM exhibits large discrepancy in proportion with the nonlinearity (convergence cannot be achieved in cubic and higher order functions). Therefore, by inserting weight matrix, the linear weighted response surface method (LWRSM) can be determined.

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Fig. 4. 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 : adaptive linear RSM in brown line; g3 : adaptive weighted linear RSM in violet line; g4 : basic quadratic RSM in yellow line; g5 : adaptive quadratic RSM in red line; g6 : adaptive weighted quadratic RSM in green line).

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 adaptive weighted response surface (AWRS) 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 linear adaptive weighted 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 The validation of the LAW-RSM has been carried out. One of examples presented by Irfan et al. [22] has been selected for the current validation. The selection of the example is motivated by the fact that the example is a highly nonlinear limit state function. The highly nonlinear limit state function is given as, g(·) = x13 + x12 x2 + x23 − 18,

(4)

where x is a random variable and g(·) represents the nonlinear limit state function. The mean and standard deviation of the considered random variables are listed in Table 5. Tables 6 and 7 present the comparisons of four limit state functions selected for the current verifications. In the comparisons, six different RSMs, i.e., linear RSM, linear weighted RSM, linear adaptive weighted RSM, quadratic RSM, quadratic

weighted RSM, and quadratic adaptive weighted RSM, have been employed to obtain the coefficients of the limit state functions. In the first three methods, the example of the nonlinear limit state function described as Eq. (5) is fitted by three coefficients, while the last three methods uses five coefficients. In order to obtain the reliability index, advanced first order second moment (FOSM) method has been employed. g(·) = MGn − (−1.81 × 107 + 1.02 × 105 x1 + 3.06 × 104 x2 + 2.23 × 104 x3 − 1.2 × 104 x12 − 0.156x22 + 0.139x32 ). (5) As described in Table 8, the reliability index by LAW-RSM shows relatively good correlation with that of MCS, being 8% discrepancy, while other five methods exhibit large differences in comparison with the MCS solution. In the MCS simulation, reliability index of 2.533 (i.e., the minimum distance from original point) is evaluated using 1.70 and 1.97 for the random variables of X1 and X2 , respectively, shown in Fig. 4. In comparison with linear RSM and quadratic RSM, the latter shows slightly improved result. This is due to the fact that the quadratic RSM has more fitting points than the linear RSM. This is particularly true since both methods are regression equations. However, reliability index discrepancy is dramatically reduced with the adaptive method and the multiplication of weighting matrix, particularly in the case of linear adaptive weighted RSM. 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

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Table 9 The converged coefficients of the responses surface function composing limit state functions of Table 7 through adaptive iterations by quadratic RSMs Run number Reliability index Probability of failure, Pf Coefficients of response surface function

1 2 3 4 5 6 7

Best fit (Eq. (2))

1 1.086 8.612E − 01 −1375.044 −75.000 100.000 −285.206 39.900 0.000 29.700

2 1.274 8.986E − 01 −2466.807 156.422 44.706 0.000 22.571 24.686 0.000

3 1.455 9.272E − 01 −2296.412 146.965 149.423 0.000 21.117 17.457 0.000

4 1.588 9.439E − 01 −2067.140 156.427 178.162 0.000 19.163 14.460 0.000

−5.502E + 02

1.768E − 04

−1.210E − 04

−1.432E − 04

Weight matrix

1 2 3 4 5 6 7

3.186 3.186 3.186 3.186 3.186 26.310 1.000

0.072 0.199 1.000 0.024 0.214 0.072 0.072

1.531 1.858 1.439 2.344 1.000 1.531 1.531

0.867 0.590 0.600 0.752 1.000 0.867 0.867

Design matrix

x1 x2

4.781 8.229

5.100 5.819

4.781 4.820

4.433 4.230

Normalized design matrix

z1 z2

−1.044 −0.334

−9.800E − 01 −8.160E − 01

−1.044E + 00 −1.016E + 00

−1.113E + 00 −1.134E + 00

5 1.688 9.543E − 01

6 1.765 9.612E − 01

7 1.827 9.661E − 01

15 1.988 9.766E − 01

1 2 3 4 5 6 7

−1887.504 166.047 190.974 0.000 17.529 12.689 0.000

−1747.595 173.533 198.450 0.000 16.244 11.451 0.000

−1636.290 179.203 203.427 0.000 15.223 10.511 0.000

−1350.547 192.312 213.376 0.000 12.644 8.221 0.000

1 2 3 4 5 6 7

1.837E − 04 0.440 0.090 1.000 0.729 0.266 0.440 0.440

2.630E − 06 2.842 1.000 7.352 1.646 4.906 2.842 2.842

1.294E − 04 0.405 0.315 1.000 0.900 0.182 0.405 0.405

−2.923E − 05 1.693 1.305 1.533 2.867 1.000 1.693 1.693

Design matrix

x1 x2

4.142 3.817

3.907 3.504

3.714 3.256

3.418 2.884

Normalized design matrix

z1 z2

−1.172 −1.217

−1.219 −1.279

−1.257 −1.329

−1.316 −1.403

Run number Reliability index Probability of failure, Pf Coefficients of response surface function

Best fit (Eq. (2)) Weight matrix

the failure point less than 20% of difference, even after 15th iterative calculations in the example. This is also attributed to the fact that the quadratic adaptive weighted RSM has still two square terms of random variables X1 and X2 . The presence of the two square terms makes the minimization onerous. The converging histories are illustrated in Tables 8 and 9 for the linear and quadratic form of adaptive weighted response surface methods respectively. The linear RSM 2D graphs (Fig. 5) show nonlinear-like curves due to the projection of lines to the X1 .X2 plane in

comparison to the figures in 3D graphs (Fig. 5). In conclusion, the LAW-RSM can approximate the performance function around the design point better than the basic (traditional) RSM or quadratic form of RSMs, as verified in those examples. 3. Risk assessment for the construction phases of a suspension bridge To investigate the uncertainties in the construction phases of a suspension bridge construction, it is necessary to determine

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

391

Fig. 5. Comparison of the converged reliability among various response surface functions in 3D graphs (g0 : the limit state function in black surface; g1 : basic linear RSM in blue surface; g2 : adaptive linear RSM in brown surface; g3 : adaptive weighted linear RSM in violet surface; g4 : basic quadratic RSM in yellow surface; g5 : adaptive quadratic RSM in red surface; g6 : adaptive weighted quadratic RSM in green surface).

Fig. 6. The elevated view of the example suspension bridge.

the critical construction stages among construction stages based on the responses of the target suspension bridge. The evaluation for the reliability of a suspension bridge system by considering an ultimate limit state for the fracture of main cables and the reliability considering a constructability limit state for a safer erection control before and during construction has been performed, described as follows. 3.1. Description of the target suspension bridge system The suspension bridge studied here is called fictitiously Hanbit Bridge, which will be one of the longest suspension bridges among five suspension bridges in Korea with the anticipated year of completion in 2010. The main span is designed to be 850 m with two side spans of 255 and 220 m each. Due to the curvature in the side span, only main span is suspended by main cables. The structural shape of the bridge system is determined by the equilibrium among the deflection of stiffening girder, and the tensile force at main cables and hangers, assuming the stress-free in the stiffening girders under dead loads when the completion of construction. The side view of the bridge and the expected completed view are shown in Figs. 6 and 2(a), respectively.

The target bridge will have a fully suspended main span supported by two portal-braced, reinforced-concrete towers. The ratio of length over bay is 52. The sag ratio of height over length is 19 . The global behavior and the local-deterioration effects of cables of “the completed bridge system” were analyzed by an FE analysis [16]. As mentioned earlier, the analyses of this study are limited to the construction stages. The main cables will be retained by steel saddles at the top of each tower to accommodate the cable geometry and to transfer their loads to the towers. Made up of highly tensile and galvanized steel wire, the two main cables were built up wire-bywire by in situ aerial spinning process. The main cables have 19 strands, composed of 380 (5.1 mm of diameter) wires each. Thus, 7200 wires are in a main cable. Main cables and hangers are modeled by catenary cable element in the initial equilibrium state analysis and in the construction stage analysis with the internal forces accounting for the geometric nonlinearity caused by the cable sag. The main cables and hangers were modeled as 148 and 98 geometric nonlinear cable elements, respectively, in the FE model. The deck cross-section is an aerodynamically shaped closed steel box girder, with fabricated steel sections measuring 19.7 m wide and 3 m deep, supported at intervals of 17.5 m length by

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T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

hangers connected to the main cables. The stiffening girders and two concrete towers are modeled by 206 space frame elements. The reinforcements are determined using a 3D FE model for the pylons and anchorage blocks under various load combinations.

construction stage considering added stiffness from the initial equilibrium state. The matrix form of finite displacement analysis method is given by the following equation [18]: {F } = {[K1] + [K2] + [K3]} · {U },

(6)

3.2. FE modeling for a construction analysis 3.2.1. Sequences of construction phases The sequence of deck placing is generally performed in two different methods as follows: (1) start erection from the center of main span to pylon, or (2) start erection from pylon, similar with the free cantilever method of prestressed concrete bridges. The erection of the target bridge girder is performed by lifting the deck units from a barge by a crane above the main cables beginning at mid-span and then by connecting the bridge sections to the hangers and to each other by temporary joints. The same construction procedure was used in Hoga Kusten bridge in Sweden. The dead load of the stiffening girder and the cable systems can be handled entirely by the main cable, provided that the main cable has a configuration coinciding with the funicular curve of the applied load. The anchorage, the towers, the cables, the cable bands, and the hangers are erected in that sequence. The bridge deck will be suspended from two main cables passing over the main towers and secured onto massive concrete anchorages at each end. The bridge deck composed of fabricated steel sections will be erected by a floating crane of 1600 tf (15,696 kN) capacity on a 6000 tf (58,860 kN) barge. Construction-hinges on top of the decks will connect the deck units to complete the bridge deck assembly. Twenty-eight stages of assembling block units will take 128 days. Virtual mockup images with FE model are shown in Table 10 to check the construction sequences, constructability, and stress analysis for each construction stage. 3.2.2. FE model Due to huge geometric deformation of main cables and stiffening girders during construction phases, the construction analysis of a suspension bridge is generally performed in a reverse order from the completed model. Therefore, the geometric nonlinear analyses for the reverse construction stages are performed by unloading from the initial equilibrium state of the completion of construction with stage-dependent boundary condition and loads. Because the pylon has enough strength to withstand the slip of saddle on the top of the pylon, there was no need for any additional safety check on the pylon section of the target bridge. The safety of pylon is examined only at the stage of its construction. The increased stiffness by the tensile forces at the cable members of the target bridge is analyzed by a linear finite displacement analysis method, which omits [K3] in Eq. (6). The internal forces of the members of bridge depend on the loads, which makes load-dependent stiffness matrix, in the finite displacement analysis. If incremental additional axial force during construction stage is large, nonlinear finite displacement analysis method is used, by the iterative calculations at each

where [K1] is a linear stiffness matrix, [K2] is a geometric stiffness matrix by initial member force, and [K3] is a geometric stiffness matrix by additional member force. The initial equilibrium state of an elastic catenary cable system is firstly evaluated and the evaluation of initial equilibrium state for a bridge is performed on the basis of initial equilibrium state of an elastic catenary cable system. Through iterations, the evaluated member forces of frame at arbitrary step are introduced as initial member forces at next step to reduce the displacements under the dead load [16]. On the assumption of free stress in stiffening girders under the dead loads at the completed state, by the reverse analysis (removing members with changing boundary condition and loads) from the completed state, the FE analysis are performed at each construction phases of the suspension bridge. Longitudinal stiffening girders, pylon, and links were modeled by four 3D space beam elements having 12 degrees of freedom per node. Boundary conditions for pylon, center stay cable, splay saddle area, and the saddle area on top of the pylon are illustrated in Fig. 7. Link shoes, restraining vertical and longitudinal transformation and rotation, are located on top of pylon. A wind shoe will control the transformation of normal direction to longitudinal direction. All transformations except rotations are restraints in the splay saddle area and anchorage blocks. Both shear key and construction-hinges are modeled by 1D spring elements, fixing all the boundary conditions except the rotation of z-axis. During the assembly of deck units, the boundary conditions vary as follows: (1) Before 25th stage: Completion of lifting all deck units, assembled by temporary construction-hinges. Two pin plate type hinges connect deck units, located at the top of deck units. (2) 26th stage: After disjointing construction-hinges, fixing all degrees of freedom by high tension bolts. (3) 27th stage: Load secondary fixed loads such as asphalts. (4) 28th stage: Completion of center-stay and the construction.

3.2.3. Loads and load combinations The loads subjected to the analysis of this study are selfweights, temperature, and wind load for each construction stage. Two combination cases are considered to generate the maximum principal stress and maximum shear stress. Load case 1 is for the maximum principal stress, which is the combination of dead load (D) and temperature load (T ), D + T . Load case 2 is for the maximum shear stress, which is the combination of dead load (D), wind load (W ) and temperature load (T ), D + W + T .

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

393

Table 10 Virtual image of construction stages (modeled by AUTOCAD Civil3D) and FE analysis model (by a FE program, RM2004 [33]) for the construction stages of the main span of the Hanbit bridge Stages

Description

1

Launch project: constructing pylons (3 year), saddles, and anchorages (30 days)

2

Build catwalk, fix storm ties, haul out tramway ropes (90 days)

3

Erection of main cables: Air spinning (compact and clamp cable for 120 days) Fix bands and hangers (90 days)

4

1st step of assembling deck units (5 days)

5

2nd to 28th step of assembling deck units (122 days)

Digital mock-up image

FE analysis model

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T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

Table 10 Continued Stages

Description

6

Remove hinges, and connect girders (170 days) Add pavements (60 days)

Digital mock-up image

FE analysis model

Fig. 7. Boundary conditions for the components with node numbers: (a) top of pylon; (b) anchorage and (c) wind shoe.

3.3. Critical construction stages Considering severe environmental condition such as “typhoon” with temporary connections between members, among the construction phases of whole members of the target bridge the most significant uncertainties in the construction are associated with the deck erection of the suspension bridge. If the constructed shape is different from the designed shape, the unexpected stress concentration could cause the shortened life of the suspension bridge. Thus the bridge needs exact careful control during the erection of deck units. 3.3.1. Tensile forces in a main cable at each construction stage It is assumed that the horizontal component cannot vary because the hangers are vertical. The vertical components increase toward the pylon as more weight is being carried. Thus, compared with the tensile force at the center of main span, the forces are larger at the top of the pylon. As shown in Fig. 8, the tensile force at a main cable increases with the increase in the assembled deck units. The maximum tensile force at the main cable is reached just before the completion of construction in

the load combinations considered. Based on the resultant tensile forces at the cables, the critical construction stages for main cables are determined to be 25th to 27th stage, at which the LAW-RSM is applied. 3.3.2. Erection control for the vertical and horizontal deflections of stiffening girders While assembling deck units, the deck section joints are left unconnected except hinges until the last unit is in position because significant displacement of the main cable takes place when the dead load is progressively increased. Calculations should therefore be made in the cable tension and deformed shape during the erection sequence to maintain tolerable limits of deformation. While the deck units are progressively assembled, the deflection of the main cable will be increased, and the deck units show gap openings at the bottom. As the erection continues, these gap openings close, but openings on the top side of the deck appear finally. This is due to the fact that the deck unit is lighter during its erection, compared to its weight in its service condition because the wearing surface is missing.

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

1.2 105 Tensile Force in Cable (kN)

7 6 Safety Factor

395

5 4 3 Pylon Center

2

1 105 8 105 6 105 4 105 Pylon Center Allowable Force

2 105

1

0 0

5

10 15 20 Construction Stages

25

0

5

10 15 20 Construction Stages

25

Fig. 8. Tensile force and safety margin at a main cable during the construction stages of assembling deck units. (a) Ratio of the tensile force at the main cable to allowable force; (b) safety margin at the main cable (allowable stress = 706 MPa, 40% of tensile strength, 1765 MPa).

4

10 9

2 Deflection (m)

Vertical Deflection (m)

3

1 0 -1

8 7 6

-2 Vertical_Min Vertical_Max

-3

5

-4

Horizontal

4 0

5

10 15 20 Construction Stages

25

30

0

5

10 15 20 25 Construction Stages

30

Fig. 9. Expected deterministic (a) vertical and (b) horizontal deflections at the center of main span (the maximum value of 2.827 m at the 26th stage and the minimum value of −3.731 m at the stage 7th, and the maximum horizontal deflection value of 9.587 m at the 21st stage).

It is clear that the huge deflections and rotations could cause not only local damage between the decks themselves but also local deformations at the connection of cable band and deck hangers as well. Furthermore, the inclined hanger due to the deformation might change the vibration shape. There could be different mechanisms of wind-induced vibration by a wake galloping of hangers, for which it may be assumed and be only tested in the normal wind direction. Based on the results of the FE analysis of construction phases depicted in Fig. 9(a), the maximum and minimum vertical deflections of the assembled deck units are observed at the 7th and 26th stage, respectively. The maximum horizontal deflection is observed at the 21st stage. Although at the 1st stage the maximum vertical deflection is shown in Fig. 9(a), it can be disregarded since there will be no contact with other deck units. The critical stages determined are evaluated in Section 3.3 for the probability of control not to exceed the tolerance limit at those critical stages.

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 the cable’s section area, the elasticity of cables, and the section area of hangers as the demand terms in limit state function. The supplying terms in limit state function are tensile strength of a cable or deck plate, and the code-specified tolerance limit for deflection control of stiffening girders. 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 code-specified criteria. Table 11 shows the statistical properties of the selected random variables.

3.4. Risk assessment for the construction stages of a suspension bridge

where i is the location of observation (L/2 and on top of pylon), t is the stage number of construction for assembling the deck units from 1st to 28th, Ac is the area of cable (mm2 ), cr is the tensile strength of a main cable (MPa), and Ti (t) is the tensile force in a main cable at each construction stage (kN).

The initial design values of random variables for composing response function use initial design matrix as the average of

3.4.1. Reliability analysis for the fracture of main cables The construction stage-dependent limit state function for the fracture of main cables is expressed in the following equation: g(i, t) = Ac cr (t) − Ti (t),

(7)

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T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

Table 11 Parameters of random variables in the reliability analysis Random variables

Index

Mean value

C.O.V.

Distribution

Reference

Area of a main cable Elasticity of cable Area of a hanger Strength of a main cable Principal stress at top and bottom plates Error limit for vertical deflections Horizontal deflections

X1 X2 X3 X4 X4 X4 X4

0.1475 m2 199,000 MPa 0.002966 m2 1765 MPa 235 MPa 0.17 m 5.33 m

0.015 0.060 0.015 0.04 0.11 0.1 0.1

Normal Normal Normal Log-normal Log-normal Normal Normal

Nowak et al. Nowak et al. Nowak et al. Ayyub [36] Nowak et al. Assumed Assumed

[35] [35] [35] [35]

C.O.V.: coefficient of variation. Table 12 Converged reliability indices (for the 25th construction stage) Run number Reliability index Probability of failure, Pf

1 Pylon 14.03 0.000E + 00

Center 15.23 0.000E + 00

4 Pylon 15.28 0.000E + 00

Center 16.61 0.000E + 00

5 Pylon 15.27 0.000E + 00

Center 16.60 0.000E + 00

Coefficients of response surface function

1 2 3 4

−5.017E + 00 9.079E − 07 2.698E − 01 1.311E − 01

5.302E + 00 2.673E − 07 2.368E − 01 1.126E − 01

4.696E + 04 7.242E + 04 −6.401E − 03 3.444E − 05

4.007E + 04 6.747E + 04 −3.176E − 03 3.135E − 05

4.794E + 04 6.944E + 04 −8.888E − 03 4.113E − 05

4.056E + 04 6.597E + 04 −4.450E − 03 3.573E − 05

Design matrix [24]

X1 X2 X3

1.475E − 01 2.695E + 05 3.270E − 03

1.475E − 01 2.734E + 05 3.270E − 03

1.485E − 01 1.965E + 05 3.270E − 03

1.486E − 01 1.974E + 05 3.270E − 03

1.484E − 01 1.955E + 05 3.270E − 03

1.486E − 01 1.968E + 05 3.270E − 03

Normalized [24]

Z1 Z2

3.625E − 12 5.901E + 00

9.032E − 13 6.233E + 00

4.454E − 01 −2.125E − 01

5.152E − 01 −1.309E − 01

4.264E − 01 −2.946E − 01

5.034E − 01 −1.833E − 01

Using the variation of random variables in Table 11, the coefficients of response surfaces are determined, after updated iteratively and converged by the LAW-RSM, to compose the limit state functions for 1st and 12th construction stages as follows: g(Pylon, 1st) = 260, 337.5 − (0.648 + 3.781 × 10−7 x1 + 0.101x2 + 0.0557x3 ),

(8)

g(Center, 1st) = 260, 337.5 − (−1.01 + 1.868 × 10−7 x1 + 0.0749x2 + 0.0375x3 ),

(9)

g(Pylon, 12th) = 260, 337.5 − (−3.165 − 4.947 × 10−7 x1 + 0.213x2 + 0.106x3 ), g(Center, 12th) = 260, 337.5 − (−3.741 + 3.155 × 10 + 0.185x2 + 0.0891x3 ).

(10) −7

x1

the 25th stage of assembling deck units, the reliability indices on the top of pylon and the center of main span is determined to be 15.27 and 16.60, respectively, shown in Table 12 with the adaptive weighted information. After constructing limit state functions, the reliability indices and the probability of failure are evaluated by FOSM method of Rackwitz–Fiessler method [34]. The converged fitted functions for the 25th, 26th and 27th construction stages are presented as follows: g(Pylon, 25th) = 260, 337.5 − (47, 940 + 69, 440x1 − 8.888 × 10−3 x2 + 4.113 × 10−5 x3 ), (12) g(Center, 25th) = 260, 337.5 − (40, 560 + 65, 970x1 − 4.450 × 10−3 x2 + 3.573 × 10−5 x3 ), (13)

(11)

The above equations are fitted functions for 1st and 12th construction stages by an LAW-RSM at the location of top of pylon (Eqs. (8) and (10)) and at the center of main span (Eqs. (9) and (10)). The supplying term, value of 260,337.5 kN in the limit state functions, is the tensile strength of a cable multiplied by the sectional area of the cable. 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%. For the example of

g(Pylon, 26th) = 260, 337.5 − (46, 000 + 75, 360x1 − 6.375 × 10−3 x2 + 3.876 × 10−5 x3 ), (14) g(Center, 26th) = 260, 337.5 − (39, 920 + 68, 020x1 − 3.610 × 10−3 x2 + 3.573 × 10−5 x3 ), (15)

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

3.4.2. Reliability analysis for the constructability of erecting deck units In the current design guideline for cable steel bridges in Korea [18], vertical deflections during construction stages are controlled based on a function of sag ratio of main cables. Excessive deformation during construction could result in plastic deformation of cable band or local damage in the stiffening girders and temporary hinges. The design for the vertical deflection of stiffening girder needs to consider errors during the fabrication and construction of main cables based on the sag of the cables as referenced [18]:

30 pylon center

Reliability Index

25

20

15

10

C =

5

0 0

5

15 10 20 Construction Stages

397

25

Fig. 10. Reliability indices of the main cable during the erection of deck units at the locations of 0 and L/2 from pylon.

g(Pylon, 27th) = 260, 337.5 − (58, 070 + 81, 420x1 + 4.203 × 10−5 x2 + 3.187 × 10−5 x3 ), (16)

850 L = = 0.17 m, 5000 5000

(18)

where L is length of center span (m). Due to the uncertainties in the temporary structures and construction methods, there is no official guideline for the tolerance limit of the horizontal deflection during construction. Therefore, for the constructability limit state during erection of deck units for the control of vertical and horizontal deflections, in this pilot test the deformation limit values for the completed structure are used as the supplying term in the limit state functions under the consideration. The construction stage-dependent constructability limit states for vertical and horizontal deflections at L/2 are presented as follows:     n  L L gvertical (stage) = + − a0 + (19) ai xi , 200 5000 i=1

g(Center, 27th) = 260, 337.5 − (52, 200 + 70, 740x1 + 2.522 × 10−5 x2 + 3.026 × 10−5 x3 ). (17) The above equations are the fitted limit state functions for 25th, 26th, and 27th construction stages by LAW-RSM at the location of top of pylon (Eqs. (12), (14) and (16)) and at the center of main span (Eqs. (13), (15) and (17)). The fitted limit state functions are used to evaluate the probability of failure and reliability indices as presented in Fig. 10. The safety of a main cable decreases continuously while deck units are loaded. As mentioned earlier, the top of pylon shows lower safety than the safety at the center of main span due to its vertical load components of the tensile forces. After adding secondary fixed loads at 27th stage, the reliability index drops to the minimum value of 13.1, which is still rather a very high value. If it is compared with the conventional safety factor, the minimum safety values are 1.487 for the top of pylon and 1.664 for the center of the main span at the 27th stage, which is hardly judged to be safe. The higher value of reliability indices (than the reliability indices of cable stayed bridges in Refs. [20,21]) is due to the low variation of the manufacturing quality for wires in the main cables as assumed 0.06 (Table 11). Therefore, if an increased coefficient of variation for the wire during construction is found, the reliability indices will be decreased. In addition, the safety level needs to be maintained over 100 years, which cannot be always guaranteed when considering unexpected loads [16].

  n  L ghorizontal (stage) = ai xi , − a0 + 150

(20)

i=1

where gvertical (stage) is the limit state function for the allowable deflection of L/200 and the tolerance limit for the error of L/5000 in the control of vertical deflections, during the construction stages, ghorizontal (stage) means the horizontal deformation limit state function for the bridge structure at each construction stage, L is the span length of center span (850 m), the values in parenthesis are the responses of bridges system as demand terms in limit state function when areas of main cables and hangers are perturbed by applying LAW-RSM. As discussed earlier, the critical construction stage for the vertical deflection of the deck units is the 7th stage. For the purpose of evaluating the probability of erection control of this stage, two MCS with LAW-RSM are performed; one for the probability of control less than 4.25 m (=L/200) and the other one for the probability of control less than 4.25 + 0.17 m (=L/5000). The results are shown in Fig. 11. For the other construction stages, only LAW-RSM is used to calculate the reliability indices. The MCS method exhibits results similar to LAW-RSM, assuming the supplying term of 4.42 m (control limit) as a constant. By considering standard deviation of 0.11 for supplying term of 4.42 m, the results of LAW-RSM show higher probability of failure values than the MCS result as manifested by the Pf of 19.5% at the 7th construction stage.

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T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400 Normal(3.94732, 0.23663)

Normal(5.43179, 0.32533)

1.8

1.4

1.6

1.2

1.4

1.0 1.2 1.0

0.8

0.8

0.6

0.6

0.4 0.4

0.2

0.2

0.0

0.0 2.5 < 2.500

3.0

3.5

4.0

4.5

97.7%

2.3%

5.0 >

4.420

Fig. 11. The expected statistical vertical deflections of the center of main span (L/2) at the 7th stage, as evaluated by 5000 MCS for the probability of exceeding 4.42 m, best fitted to normal standard distribution as presented in probability density function and cumulative distribution function form, with Pf = 2.3% ( = 1.99).

3.4.3. Reliability analysis for the horizontal deflections of deck units MCS, LAW-RSM, and FOSM are used to compute the reliability indices for the 25th construction stage, summarized in Fig. 12 and Table 12. The results of MCS method manifest smaller probability of exceeding the tolerance value of deflection control than the results of FOSM by the same reasoning of the simulation case for the vertical deflection as discussed in Section 3.3.2. As shown in Table 12, using the tolerance of completed structure, the horizontal deflection cannot be satisfied until fixing all the degree of freedom among deck units at 26th erecting stage of stiffening girders. After 21st stage, due to the increased stiffness of main cables and the self-weights of the girders, the horizontal deflection will be decreased. If measuring and protecting are appropriate for a local damaging at the connections of hangers, cable bands, and buffering device between deck units from the severe lateral wind loads, there might be no significant risks during the construction of superstructures of Hanbit bridge.

4.0

4.5

5.0

<

5.5

6.0

76.5%

4.000

6.5

7.0 >

23.5%

5.766

Fig. 12. Expected statistical horizontal deflections of the center of main span at the 25th stage, as evaluated by 5000 MCS, probability of exceeding 5.667 m, best fitted to normal standard distribution as presented in probability density function and cumulative distribution function form, with Pf = 23.5% ( = 0.72).

30 25

Pylon_Cable

Vert_deflection

Center_Cable

Horiz_deflection

20 15 10 5 0 -5 0

5

10 15 20 Construction Stages

25

Fig. 13. The changing reliability indices for main cable and deflections of stiffening girders during the construction of suspension bridge.

3.5. Reliability of the Hanbit bridge by each construction stage The investigated reliability indices during the erection of deck units of Hanbit bridge are presented in Fig. 13 and Table 13. The results of reliability indices for stresses and

deflections indicate that the target suspension bridge manifests clearly huge geometric nonlinear deformation during construction.

T. Cho, T.S. Kim / Finite Elements in Analysis and Design 44 (2008) 383 – 400

399

Table 13 Reliability indices for main cable and deflections of stiffening girders during the construction of suspension bridge Component

Location

Cable

Pylon Center of span

Deflection

Construction stages Vertical Horizontal

Stresses

Tensile Tensile

Construction stages 1

7

14

25

26

27

23.15 25.9

19.93 21.62

16.22 17.5

15.27 16.6

15.37 16.64

13.1 14.23

1 4.87 −2.94

7 0.86 −3.65

14 6.39 −3.75

21 6.72 −3.85

25 6.01 −0.21

27 13.53 1.35

4. Concluding remarks As providing computer aided integrated solution the decision making system VA, will allow the efficiency of collaboration, increased design quality, and improved knowledge based concurrent engineering, which will cost very high by the limited expert in future As a part of VA project in Korea, VA-Risk aims to develop a probabilistic risk assessment system, which will help our decision for the sequence and method of structures before construction. Working interactively with VA modules, VA-Risk will provide the better communication, enhancing collaboration, interoperability in 3D design and construction, reusability of database, and the efficiency of alter analysis. Before the completion of construction, many civil-infrastructures encounter critical loads. Some of them already collapsed, and many of them may still have initial imperfections, caused by errors in planning, design, manufacturing of materials, or excessive loads before the completion of construction. In this study, quantification and the evaluations of uncertainties in construction phases of a suspension bridge have been investigated. Based on the results of the reliability analyses, the following conclusions are drawn: (1) The critical construction phases were determined from the result of FE analysis. The 25th, 26th, and 27th construction stages were investigated to assess the probability of rupture of a main cable. (2) For the evaluation of constructed response surfaces of limit state functions, the analyzed results have been compared with the conventional safety indices and the allowable errors in erection control during the critical construction stages. Regarding the fracture limit state of main cables during constructions, the minimum reliability index was 13.1, which is still a very reliable state. However, it would be too hasty to jump to the conclusion since the conventional safety index of 1.487 is at this critical stage, thus the safety will depend on the strength quality of the steel wires in main cable, which was assumed with the coefficient of variation as 0.04. (3) Erection control for the vertical and horizontal deflections during the critical construction stages of 7th and 21st are considered based on the tolerance for the cable sag ratio and the serviceability limit state of the constructed long span bridge against vertical deformation. The target suspension bridge shows typical geometric nonlinear

deformation during construction. The structure might appear to be safe based on the conventional stress analysis, but there can be secondary local deformation under severe environmental loads. (4) Based on the tolerance values for the structure being constructed, excessive horizontal deflection could result in a local damage at the connections of hangers, cable bands, and buffering device between deck units (stiffening girders) by a severe lateral wind load during the 26th erecting stage of deck units, when all the degrees of freedom among deck units are fixed. Acknowledgments This work was supported by the research grant from MOCT (Ministry of Construction and Technology) of Korean Government (Grant no.: 06HIGH-TECH FUSION-E01) and Sustainable Building Research Center of Hanyang University which was supported by the SRC/ERC program of MOST (Grant R112005-056-01003). The 3-dimensional graphic illustrations are supported by the general manager, Han, in SR Partners Inc. The authors hereby express their sincere appreciation. References [1] PMIS Solutions in the Korean Construction Industry, The construction and economy research in statute of Korea (CERIK), Research Report, 2003. [2] S.-H. Han, K.-H. Chin, M.-J. Chae, Evaluation of CITIS as a collaborative virtual organization for construction project management, Autom. Constr. 16 (2007) 199–211. [3] Korea Rail Network Authority, Problems and solutions for the safety managements in the construction of railway structures, Research Report 05-2-394, 2006. [4] W.G. Cochran, Sampling Techniques, third ed., Wiley, New York, 1977. [5] G. Fishman, Monte Carlo, Concepts, Algorithms, and Applications, Springer, Berlin, 1995. [6] J. Cheng, M.J. Druzdzel, An adaptive importance sampling algorithm for evidential reasoning in large Bayesian networks, J. Artif. Intell. 13 (2000) 155–188. [7] B. Minasny, A.B. McBratney, A conditioned Latin hypercube method for sampling in the presence of ancillary information, Comput. Geosci. 32 (9) (2006) 1378–1388. [8] D. Novák, D. Lehký, ANN inverse analysis based on stochastic smallsample training set simulation, Eng. Appl. Artif. Intell. 19 (7) (2006) 731–740. [9] J.S. Rosenthal, Markov chain convergence: from finite to infinite, Stochastic Process. Appl. 62 (1996) 55–72.

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