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International Conference on Recent Advances in Railway Engineering (ICRARE-2013) Iran university of science and Technology β Tehran β I.R. Iran βApr 30, May 1, 2013
Optimization of Post-Tensioned Concrete Box Girder Double-Track Railway Bridges Using Genetic Algorithm M. ARAB NAEINI1, A. KHEYRODDIN2, H. NADERPOUR3, R. ARAB NAEINI4 1
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Graduate Student, Semnan University;
[email protected] 2 Professor, Semnan University;
[email protected] 3 Assistant Professor, Semnan University;
[email protected] Undergraduate Student, Islamic Azad University Shahreqods Branch;
[email protected]
Abstract In the design of bridge superstructure, "weight" is an important factor especially in long spans; because reducing weight will reduce seismic forces and cost of superstructure and substructure. Among the prestressed concrete bridges, posttensioned concrete box girder is one of the most common and suitable bridges. Different variables are involved in the design of this type of the bridges. This matter leads to a numerous designs for a particular bridge span length and deck width. So, obtaining the optimum solution by conventional methods is a very difficult and time-consuming work. In this study, post-tensioned prestressed concrete box girder double-track railway bridge superstructure is optimized using Genetic Algorithm (GA) by considering several variables including cross sectional geometry, number of tendons, number of strands per tendon and tendons configuration. Explicit and implicit constraints are formulated according to practical and functional requirements based on AASHTO standard specifications. GA is capable to locate the global minimum in a short time. In this way, optimal solution will be achieved by minimum human effort. Keywords: Optimization, Prestressed Concrete, Railway Bridge, Box Girder, Genetic Algorithm (GA). 1 Introduction Among the concrete bridge deck sections, single-cell box girder is one of the most common sections for railway bridges. Single box arrangements are efficient for both longitudinal and transverse designs, and they produce an economic solution for most medium and long span bridges [1]. In the design of prestressed post-tensioned concrete box girder bridges, several variables are involved that are related to each other. This matter leads to a variety designs for a particular span length and width which obtaining optimum solution is difficult. Optimization by using new algorithms like Genetic Algorithm (GA) provides a suitable tool for finding feasible minimum. GA is an evolutionary algorithm that is capable to locate the minimum of a function in a short time. In the design of bridge superstructure, "weight" is an important factor especially in long spans; because reducing weight will reduce seismic forces and cost of superstructure and substructure. During the past decades, a lot of research has been done in the area of structural optimization. As one of the first work on optimization of concrete bridges, Torres et al. (1966) optimized the prestressed concrete bridges by using a linear programming method. Yu et al. (1986) used general geometric programming in optimization of a prestressed concrete box girder in a balanced cantilever bridge. Previous procedure was also used by Barr et al. (1989) for minimization of a continuous three-span bridge RC slab. Lounis and Cohn (1993) presented the optimum design of RC slabs on precast, post-tensioned concrete girders. Cohn and Lounis (1994) applied their optimization approach to multiobjective optimization of partially and fully prestressed concrete bridges. Lounis and Cohn (1995) also concluded that voided slab decks are more economical than box girders for short spans (less that 20 m) and wide decks (greater than 12 m), and single-cell box girders are more economical for medium spans (more than 20 m) and narrow decks (less than 12 m). Fereig (1996) presented the optimum preliminary design of single span bridges consisting of cast-in-place RC deck and girders by the Simplex method. Ayvaz and Aydin (2009) studied the optimization of a pre-tensioned I-girder bridge using GA. Ahsan et al. (2012) presented the optimum design for post-tensioned I-girder bridge using EVOP (Ghani 1989). In this study, weight optimization of simply supported, post-tensioned cast in place (CIP) concrete box girder railway bridge superstructures is investigated. Variables considered are cross sectional dimensions of the girder, tendons configuration, number of tendons and number of strands per tendon. Constraints on these variables are extracted from AASHTO standard specifications and practical restrictions. Loading the bridge is based on "Standard Loads for Bridges" of Iran. Finally, after formulation of analysis and design of the bridge according to AASHTO, GA is applied for optimization.
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Yazdani, S. and Singh, A. (Eds.)
2 2.1
Problem Formulation Bridge Live Loads
Loading the railway bridge is according to "Standards Loads for Bridges" of Iran. Train live loads on the bridge are shown in figure 1. Pedestrian live load is 4 KN per square meter. Impact factors for shear and moment are also considered according to code formulas.
Figure 1: Train live load on railway bridges
2.2
Design Variables and Constant Parameters
Several different variables considered in this study are cross-sectional dimensions of the box girder, number of tendons in each web, number of strands per tendon, tendons arrangement, slabs reinforcements and prestressing force. These variables are tabulated in table 1. A typical cross section of box girder with some of the variables is shown in figure 2.
Figure 2: Box girder cross section of double-track railway bridge Table 1: Design Variables and Explicit Constraints
No.
Variables
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Girder depth (m) Top slab thickness (cm) Bottom slab thickness (cm) Web thickness (cm) Length of cantilever (m) End thickness of cantilever (cm) Beginning thickness of cantilever (cm) Length of haunch (cm) width of haunch (cm) Number of strand per tendon Number of tendons in each web Number of anchorage in each row Lowest anchorage position (cm) Prestressing force (% of ) Top slab reinforcement ratio Cantilever slab reinforcement ratio
Symbol
Type Continuous Continuous Continuous Continuous Continuous Continuous Continuous Continuous Continuous Integer Integer Integer Continuous Continuous Continuous Continuous
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Explicit constraint
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New Developments in Structural Engineering and Construction
Constant design parameters are bridge length and width, material properties, anchorage system and live loads. Freyssinet C-range is used for anchorage system and 15 mm seven wire strands are used for tendons (see Table 2). Table 2: Constant design parameters
Constant parameters
values
Span length (L) Bridge width (W) Concrete strength ( ) Concrete strength at transfer ( ) Tensile strength of prestressing steel ( ) Yield strength of prestressing steel ( ) Yield strength of reinforcement ( ) Ballast thickness Unit weight of reinforced concrete ( ) 2.3
30 m 12 m 40 MPa 0.7 1860 0.9 40 0.5 m 2500
Objective Function
The goal of this optimization is to minimize the weight of post-tensioned prestressed concrete box girder double-Track railway bridge superstructure. The objective function is defined as follow: (1) Where girder. 2.4
=weight of the bridge superstructure per square meter of the deck and
= cross sectional area of the
Explicit Constraints
Explicit constraints are lower and upper limits on design variables which are based on construction restrictions. These constraints are summarized in table 1. Maximum length of cantilever is assumed 3 meter, because AASHTO limited cantilever length to 1/4 of bridge width (W). Prestressing force is considered not less than for optimum use of prestressing steels and upper limit is according to AASHTO. 2.5
Implicit Constraints
These constraints control the performance requirements of the bridge. Implicit constrains are formulated according to AASHTO (2002). The bridge is designed in both longitudinal and transverse directions. In longitudinal direction, dealing with a prestressed concrete design and in transverse direction, dealing with a reinforced concrete design. Totally, 105 implicit constraints are considered in this study that explained in follow. 2.5.1
Allowable Stress Design Constraints
These constraints that control the stresses in top and bottom concrete fibers to remain in allowable tensile and compressive stresses are as follow: (2) (3) where , and = working stress, allowable compressive stress and allowable Tensile stress respectively. F= prestressing force; A= cross-sectional area of the girder; e = tendons eccentricity; M= working moment; S= section modulus of the girder. Five critical sections are considered along the girder span: (1) section at midspan (2) section at the end of transition zone (assumed 1.5h according to AASHTO) (3) section after the anchor set (4) section at the end of diaphragm (assumed 0.8 m) (5) End section
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Yazdani, S. and Singh, A. (Eds.)
Two stages of loading, transfer of prestressing force and service conditions are considered. Instead of using a lumped-sum prestressing losses, both instantaneous and longtime are calculated by AASHTO formulas for greater precision.
2.5.2
Allowable Stress in Prestressing Steel Constraints
These constraints control tensile stresses in pressing steel at two sections and two stages according to AASHTO as follow: (4) (5) (6) Where , = prestressing force at sections 5 and 3 after instantaneous losses; after longtime losses; = prestressing steel area. 2.5.3
= prestressing force at section 3
Ultimate Flexural Strength Constraints
These constraints are according to Ultimate Strength Design (USD) method, which considered at sections 1 to 4 and are as follow: (7) Where 2.5.4
= ultimate bending moments at different sections;
= flexural strength of the section.
Minimum and Maximum Prestressing Steel Constraints
Minimum prestressing steel constraints considered at sections 1 to 4, are as follow: (8) Where = cracking moment. Maximum prestressing steel constraints considered at sections 1 to 4, are as follow: π π Where π= reinforcement index; π = upper limit of reinforcement index. 2.5.5
(9)
Deflection Constraints
Deflection of the girder by considering creep effect should be limited as follow: (10) Where = longtime deflection in midspan; L= span length of the bridge. 2.5.6
Ultimate Shear Strength Constraints
These constraints that control shear stresses at different sections are as follow: (11) Where = ultimate shear; =strength reduction factor for shear; = nominal shear strength provided by concrete and shear reinforcement respectively. Shear strength is controlled at sections 1 to 4. Also, section at 0.25h, which "h" is depth of the girder, is considered to compute shear reinforcement. 2.5.7
Slabs Design Constraints
Three slabs including top, bottom and cantilever slabs are designed according to ultimate strength design method as Eq. 7. Bottom slab is under self-weight only. Loads on top slab are dead load including weight of slab, ballast and rails and two concentrated live loads of 25/2 ton that are distributed by 2:1 slope as shown in figure 3. Cantilever slab resists dead loads including self-weight, pavement and curb and live loads including pedestrian live load and probable concentrated live load of train depending on the variable of cantilever length.
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New Developments in Structural Engineering and Construction
Figure 3: Distribution of concentrated train loads
2.5.8
Cantilever Slab Deflection Constraint
Deflection of cantilever slab under loads that mentioned above is limited as follow: (12) Where = deflection at the end of cantilever, 3
= length of cantilever.
Optimization Process
The optimization problem is determined by defining 16 variables, 32 explicit constraints, 105 implicit constraint and objective function. Numerous variables and constraints are considered in this study. Objective function and most of constraints are nonlinear function of design variables. Many mathematical linear and nonlinear programming methods have been developed for solving optimization problems during the last three decades. Some methods, such as the penalty function method search for a local optimum by moving in a direction related to the local gradient. For the optimum design of large structures, these methods become inefficient due to a large amount of gradient calculations.(Adeli 2006) Therefore, a powerful optimization algorithm is needed so that locates the global minimum in a short time without be entrapped in local minima. Genetic algorithm is used for optimization in this study, which is optimization technique based on Darwinβs theory of evolution and the natural law of the survival of the fittest. GA is a global search procedure for gradually improving the solution in succeeding populations using operations that mimic those of the natural evolution such as reproduction, crossover, and mutation and performs a random information exchange to create superior offsprings. (Adeli 2006) different phases of GA are shown in figure 4.
Figure 4: Phases of genetic algorithm
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Yazdani, S. and Singh, A. (Eds.)
Some advantages of GA are: ο· Global Search Method: GA search for the function optimum starting from a population of points, not a single one. This characteristic suggests that GA is global search method that can reduce the probability of finding local minima, which is one of the drawbacks of traditional optimization methods. ο· Blind Search Method: GA only uses the information about the objective function and doesn't require knowledge of the first derivative or any other auxiliary information, allowing a number of problems to be solved without the need to formulate restrictive assumptions. For this reason, GA is often called blind search method. 4
Optimization Results
Optimization of the post-tensioned concrete box girder double-track railway bridge is performed using genetic algorithm. The optimum design is obtained in a short time. Weight of the optimum design and related variables are tabulated in table 3. The optimum bridge cross-section is also shown in figure 5. As seen in table 3, the weight of the optimum bridge design is 1.233 ton per square meter of the deck. Cantilever length is obtained maximum value so that webs rely under tracks and hence reduce stresses in top slab. Totally, 190 seven wire strands are used for whole section. Prestressing force is obtained , which isn't reached to its maximum value , because stress in prestressing steel constraints won't be satisfied.
Figure 5: Optimum post-tensioned concrete box girder double-track railway bridge cross section Table 3: Optimum bridge design weight and related variables
No.
Variables
Symbol
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Girder depth (m) Top slab thickness (cm) Bottom slab thickness (cm) Web thickness (cm) Length of cantilever (m) End thickness of cantilever (cm) Beginning thickness of cantilever (cm) Length of haunch (cm) width of haunch (cm) Number of strand per tendon Number of tendons in each web Number of anchorage in each row Lowest anchorage position (cm) Prestressing force (% of ) Top slab reinforcement ratio Cantilever slab reinforcement ratio Weight per square meter of the deck (
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)
Type
Optimum Design
Continuous Continuous Continuous Continuous Continuous Continuous Continuous Continuous Continuous Integer Integer Integer Continuous Continuous Continuous Continuous
3.3 27 17.5 29 3.00 17.5 22 136 25 19 5 1 97 81 0.028 0.019
1.233
New Developments in Structural Engineering and Construction
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Conclusions
Optimization of post-tensioned concrete box girder double-track railway bridge superstructures was presented. 16 different variables and a total of 137 constraints are considered based on AASHTO (2002) and construction restrictions. Objective function was the weight of bridge superstructure per square meter of the deck. Prestress losses calculated by code formulas for greater precision, instead of using a lumped-sum value. After formulation of analysis and design of the bridge in longitudinal and transverse directions, genetic algorithm is used for optimization and optimum design is obtained in a short time. GA is a powerful algorithm that is capable to locate the global optimum of complex problems with a large number of variables and constraints. It was observed that optimum design weighs 1.233 ton per square meter of the deck for a double-track railway bridge by 30 meter span length. It is concluded that by availability of fast computer and powerful algorithms like GA, time is appropriate to perform optimization on large structures such as bridges. In this way, by one time formulation the bridge analysis and design, it is possible to optimize a bridge with desired characteristics. Therefore, in addition to saving in materials, time, cost and human effort, preservation of the environment is resulted. References [1] Hewson N. R., Prestressed Concrete Bridges: Design And Construction, Thomas Telford, 2003. [2] Torres, G. G. B., Brotchie, J. F., and Cornell, C. A., A Program For The Optimum Design of Prestressed Concrete Highway Bridges, PCI J., 11(3), 63β71, 1966. [3] Yu, C. H., Das Gupta, N. C. and Paul, H., Optimization of Prestressed Concrete Bridge Girders, Eng. Optim., 10(1), 13β24, 1986. [4] Barr, A. S., Sarin, S. C. and Bishara, A. G., Procedure for structural optimization. ACI Struct. J., 86(5), 524β531, 1989. [5] Lounis, Z. and Cohn, M. Z., Optimization of Precast Prestressed Concrete Bridge Girder Systems, PCI J., 38(4), 60β 78, 1993. [6] Cohn, M. Z. and Lounis, Z., Optimal Design of Structural Concrete Bridge Systems, J. Struct. Eng., ASCE, 120(9), 2653β2674, 1994. [7] Lounis, Z. and Cohn, M. Z., Computer Aided Design of Prestressed Concrete Cellular Bridge Decks, Microcomputer in Civ. Eng., 10(1), 1β11, 1995a. [8] Fereig, S. M., Economic Preliminary Design of Bridges With Prestressed I-Girders, J. Bridge Eng., 1(1), 18β25, 1996. [9] Ayvaz, Y. and Aydin, Z., Optimum Topology And Shape Design of Prestressed Concrete Bridge Girders Using A Genetic Algorithm, Struct. Multi. Optim., 41(1), 151β162, 2009. [10] Ahsan, R., Rana, S., and Ghani, S., Cost Optimum Design of Posttensioned I-Girder Bridge Using Global Optimization Algorithm, J. Struct. Eng., 138(2), 273β284, 2012. [11] Ghani, S. N., A Versatile Algorithm for Optimization of A Nonlinear Non-Differentiable Constrained Objective Function, UKAEA Harwell Rep. No. R-13714, HMSO Publications Centre, London, 1989. [12] Management and Planning Organization of Iran, "standards loads for bridges", Publication No. 139, First revision, 2000 [13] AASHTO, Standard Specifications for Highway Bridges, 17th Ed., Washington, DC, 2002. [14] Freyssinet Inc., The C Range post-tensioning system, May, 2010. [15] Sarma, K. C. and Adeli, H., Cost Optimization of Concrete Structures, J. Struct. Eng., ASCE, 124(5), 570β578, May, 1998. [16] Adeli, H., Sarma, K. C., Cost Optimization of Structures: Fuzzy Logic, Genetic Algorithms, and Parallel Computing, John Wiley & Sons, 2006.
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