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6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

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GA based Optimal Design of Steel Truss Bridge

R. Pandia Raj1, V. Kalyanaraman2

1 Research Student, University of Cambridge, Cambridge, United Kingdom, E-mail: [email protected] 2 Professor, Indian Institute of Technology Madras, Chennai, India, E-mail: [email protected]

AbstractThe economy of the steel truss girder bridge superstructure, used extensively in medium and long span ranges, is affected by many

factors such as the cost of material, dictated by the configuration and depth of truss, shape and size of members, and cost offabrication. Most conventional methods of optimization deal with only member optimization assuming only continuous designvariables, leading to designs mathematically feasible, but not necessarily practical. Genetic Algorithms (GA) can deal with mixed

design variable problem, complex design domain requirement, complex constraint equations as well as objective function and hencelead to optimal feasible designs. The objective of the paper is to demonstrate the efficient integration of GA for optimization with a

software in object-oriented environment to design large scale truss bridge superstructure for railway loadings. The GA has beenlinked to a structural design module, consisting of design rules (Indian Railway Standard and Bridge Rules), design methodology and

Finite Element Analysis (FEA). The objective function to be minimized is the total costs of the structural system. The paper gives thedetails of the software and discusses the results obtained.

Key Words Computer-aided integrated design, Genetic algorithms, Object-oriented analysis and design, Optimization, Steel trussgirder bridge.

1. IntroductionThe desire to improve a design without compromising the structural integrity has been a strong driving force behind the development

of various optimum design methods. Large volume of optimal structural design based on many optimization techniques has beencarried out in civil, mechanical, electrical, aerospace and other engineering fields. But the research reported in last three to fourdecades have not had much impact in the field of civil engineering due to the following main reasons:

• Majority of the optimal design methods deal with only for member size or configuration optimization not considering varioustopology, continuous and discrete variables in the design etc.

• Many optimal design methods based on numerical algorithms, use gradient information on response quantities as well asconstraints functions. Convergence is checked using this information, often leading to a local optimal solution or numericalinstability

• Often optimum design may not be practical, since it has been difficult to represent such practical constraints in most typical

optimization techniques. • The mathematical complexity of the numerical optimization algorithms is another reason. In order to simplify the problem onlyweight of the structure is often considered in the objective function, disregarding the effect of other factors, such as cost offabrication etc.

These observations point to the direction in which research and development activities in structural optimization should proceed, toevolve methodologies useful for the engineering design profession, and at the same time bring about cost effective designs.

Steel truss girders have been used successfully in the intermediate and long span railway and highway bridges, since the internalforces in the members are essentially only tensile or compressive. However mathematical representation of the design problem is

complex. The economy of the steel truss girder bridge superstructure is affected by many factors such as cost of fabrication, inaddition to weight of material used. Further, strength of compression members depends upon its slenderness ratio. The members may be made up of standard rolled sections, built up using rolled sections with plates or using only plates. The design parameters defining

member dimensions (breadth, depth and thickness of the elements of the cross section) configuration and topology are a combinationof continuous and discrete variables. Most of conventional methods of optimization can not deal with the complexity of the practical

optimum design problem leading to only mathematically feasible but not practical solutions. By using GA, which can deal with themixed design variable problem, complex design domain requirement as well as complex constraint equations and objective function,one can arrive at optimal feasible designs both from the mathematical as well as the practical point of view. This paper demonstrates

use of GA for solving optimum design of steel truss girder superstructure design. The GA based optimization is in an object-oriented design paradigm. Details of GA and OOM implementation are reviewed and a sample result is presented.

2. Structural Optimization using Genetic AlgorithmsThe basic principles of GA were presented by Goldberg [4]. Jenkins [5] demonstrated structural optimization using the geneticalgorithm. Rajeev et al [11] worked on discrete optimization of structures using genetic algorithms. Adeli [1] integrated genetic

algorithm for optimization of space structures. The layout and sizing optimization of a typical steel roof using GAs was done byKoumousis et al [6]. Ohsaki [9] proposed a global search algorithm for topology optimization of trusses based on the genetic

algorithms. Galante [3] applied GAs to optimize real-world trusses and transmission line towers. Rajeev et al [12] presentedimproved two phase method and variable string length genetic algorithm (VGA) for size, configuration and topology optimization of

truss bridges. Design optimization of reinforced concrete plane frames using genetic algorithm–based methodology was presented byRajeev [13]. Erbatur et al. [2] used genetic algorithms as the optimizer for the discrete optimal design of planar and space structures.

Krishnamoorthy [7] illustrated typical applications of GAs to practical design of structural systems such as steel trusses, towers,

bridges, reinforced concrete frames, shells and layout planning of buildings. The object-oriented design and implementation of a core


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genetic algorithm library consisting of all of the genetic operators with an interface to a generic objective function were discussed byKrishnamoorthy et al. [8]. Renner [10] gave an overview of applications of genetic algorithms to different domains of engineeringdesign. Sivakumar et al. [14] presented a new approach for the optimization of steel lattice towers by combining genetic algorithms

and an object-oriented approach.Literature clearly indicates that Genetic Algorithms have proved to be one of the most efficient and robust optimization

techniques for single as well as multi-objective optimization problems. The objective of almost all optimization studies is tominimize the cost. None of the studies consider the truss girder configurations as design variables and all the other complexities of

truss girder design mentioned earlier. Genetic Algorithms, which can deal with combination of continuous and discrete designvariables, can be used to achieve the objective very effectively and efficiently. Hence in the present study, which comprises theoptimization of steel truss girder superstructures on railway bridges, Genetic Algorithms has been chosen as the optimization tool.

The problem of system optimization requires a consideration of elements of the system, all design variables and constraintequations simultaneously. Whereas, the Object-Oriented Methodology represents the design space as a collection of nearlyindependent object modules. The difficulty in reconciling with these two extremes might be a reason for the existence of a few

applications of GAs based optimization procedures in an Object-Oriented environment. In the present work, details of a user-friendlysoftware for structural optimization of steel truss bridge superstructure in an integrated GAs and object-oriented environment is

presented. This software can be used to perform both conventional and optimal design of steel bridge girder superstructure.

3. Simple Genetic Algorithms Genetic algorithms, based on the mechanics of natural selection and natural genetics, combine Darwin’s theory of evolution based on

survival of the fittest and a systematic information exchange guided by random operators to form a robust search procedure. Theysimulate, artificially, the concept of survival of the fittest, wherein the genetic operators abstracted from nature form the basis of a

search mechanism, which is indifferent to the complex nature of the search space. A simple genetic algorithm (SGA) that has beensuccessfully used in many applications, illustrates the concepts and working of a typical genetic algorithm. SGA uses three genetic

operators: reproduction, crossover and mutation.The simple genetic algorithms are flexible, robust and stable. Complexity of a problem does not affect the optimization process

in the codified SGA space. While most other methods of optimization process a single point in the search space, genetic algorithms pursue a population of potential solutions simultaneously in the search space of complicated structure thus exploiting a built-intendency to find global optima. SGA can also be used beyond parameter optimization, for creative topology generation.

3.1 Transforming the Phenospace to Genospace GAs work in a different world, created through a coding scheme in which the different variables are coded (generally to binary form)and concatenated to form one long string, representing a point in the design space. The genetic operations are done on this codedstring, which acts as the medium for the structured information exchange. There are two different cases for coding a variable,

depending on whether the variable is continuous or discrete. If a variable is coded into a binary string of length n, then a total of N =2n variations of the variable are possible. The value of a continuous variable, W, can be decoded as:

S DW W

L

×+= (1)Where, WL is the lower bound on the variable W, D is the integer equivalent of the variable in the binary string and S the step size forvariable W defined as:

( ) N

W W S

LU −= (2)

The value of the discrete design variables, corresponding to integer D in the binary string is obtained as the D th value in the orderedset (S).

{ }1210 .......,.........,.........,, −= N D S S S S S S (3)

3.2 Fitness Determination The fitness of a chromosome reflects its ability to survive and reproduce. In the case of maximization problems, the fitness must be

directly related to the objective function value and in the case of minimization problems, the fitness must be inversely related to theobjective function value.

GAs search unconstrained objective function space. Many practical problems contain one or more constraints that must besatisfied. The constrained optimization problem is transformed to an unconstrained problem by associating a penalty function with allconstraint violations to obtain an augmented objective function ( f aug) and fitness (F i) as given below

( )vaug C k C f ×+= 1 (4)

augi f f f F −+= minmax (5)

Where C = Objective function; K = penalty factor; Cv = the sum of values of all violated constraints; f max, f min = maximum, minimum

value of augmented objective function in a generation.

3.3 Convergence CheckingIn an ideal situation, all the individuals in the population will attain the same best fitness at convergence. But it is very difficult to

achieve this in practice. If a specified percentage of individuals attain the same or nearly the same best fitness value, then the geneticsearch can be terminated. It means that the character patterns in the specified number of strings have to be the same. Generally, avalue of 70 to 90 % of the population reaching the best fitness is assumed as convergence. The best solution of each generation is

compared with already stored best solution up to the previous generation. If it is found to be more fit, then the present best solution

replaces the stored best solution. Once convergence is reached, the best solution is printed to a file as the optimal solution.


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3.4 Genetic Operators A simple genetic algorithm that yields good results in many practical problems is composed of three operators: reproduction,

crossover and mutation. We work with a fixed population size, which is determined by the size of the problem and the desireddiversity. The members of the first generation population are randomly initialized within the bounds of the design space and from

there on, the genetic operations are carried out on the population. Reproduction: Reproduction operation is the process by which, the rules of survival of the fittest are employed so that fit members of

the population have the greatest probability to combine to generate the next generation, while the unfit members die away. There aredifferent methods to select members of the population combining to contribute to the next generation namely: roulette wheelselection, stochastic remainder with replacement selection, and stochastic remainder without replacement selection, deterministic

selection, tournament selection and part sum selection procedure.Crossover: It is done so as to mix good genetic components that may be spread across different members of the population togenerate possibly more fit members of the next generation. A pair of individuals is selected randomly to form matches for crossover.

Cross-sites are selected along the length of the strings representing the individuals. Crossover is performed by swapping sub-stringsdefined by the cross-sites, which generates offsprings (Fig. 1). The swapping is done based on the prespecified probability of

crossover (pc). Mutation: The primary objective of mutation (Fig. 2) is to keep the diversity in the population. It consists of alteration of character

values in individual strings based on prespecified probability (pm).

Parents: 111111111111000000000000 Before

Crossover: 111111111111 101110010110000000000000 101110110110

AfterChildren: 111111110000

000000001111

Fig. 1 Crossover Fig. 2 Mutation

4. System Description

The superstructure of through type of steel truss railway bridge (Fig. 3) consists of two planes of trusses that are separated to carrysingle railway lines. The rail track is laid on sleepers supported by simply supported stringers. The stringers are connected to the

cross girders. The main truss supports the cross girders at the level of the bottom chord at the nodal point. The top and bottom chordare designed to resist resultant bending moment and the diagonals to resist the shear at the section. The two planes of trusses areconnected through cross girders and bracing at the top and bottom. The top lateral bracing along with the top chords of main trusses

form a horizontal truss system supported by end portals, which carries wind load. In addition to transmitting the wind loads, the toplateral system also serves as lateral support to compression (top) chords against out of vertical plane buckling. There are horizontal

bracings at the bottom chord level that transfer wind loads, transverse racking, longitudinal braking and tracking forces due to therailway traffic.

Fig. 3 Skeleton of a Through Type Steel Truss Railway Bridge Superstructure

The end and intermediate portal bracing is provided in through bridges in the plane of the end posts and intermediate plane. The

sway bracing is generally placed in each panel and made as deep as the clearance will allow. The function of the sway bracing is tostiffen the structure laterally and to maintain a rectangular cross-section. The connections in bridge trusses occur at the truss nodes


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where the web members are connected to the chord members concentrically. This connection usually incorporates also splice in thechord members. Further, the cross girder and the bracing systems also connect to the truss at the junction. Site connections can bemade by riveting or bolting using High Strength Friction Grip (HSFG) bolts or welding. Good site welds are difficult to achieve

where access is difficult and fatigue life of welded joints is lower than that of riveted or HSFG bolted joints. Indian Railways usesonly riveted joints and hence only riveted site connection between members is included in this study. Generally trusses have rigid

joints due to many rivets in the joint. Computer analysis can account for the joint stiffness and the consequent secondary moments bytreating the joints as rigid. The design for primary axial loads and the secondary moments is done using suitable interaction formulae.

The major loads and forces to be considered while designing a steel truss girder bridge superstructure include dead load, live load,dynamic effects, forces due to curvature and eccentricity of track, longitudinal force, racking force, wind load, and earthquake forces.The working stress method currently used in the Indian railways is followed as the design methodology.

5. Finite Element Analysis of Steel Truss Rail BridgesA finite element modeling and analysis software is exclusively developed, for the analysis of steel truss girder railway bridges by

modifying the PASSFEM software. All the elements in the truss system are modeled as beam elements, so that secondary momentsdue to rigid joints, lateral bending of verticals due to lateral loads are also evaluated in the analysis. Truss girder model is analyzed

for various load combinations and the governing design forces and moments are listed in the output file for designing the entire trusssystem.

6. Bridge Design System (BDS)The Bridge Design System (BDS), developed in the present work, is an integrated software capable of analysis, design and designdocumentation of steel truss railway bridges. Schematic of the software (BDS) is shown in Fig. 4. It can handle both conventional as

well as optimal design of bridge superstructures, according to the desire of the user. The BDS is composed mainly of fourcomponents, namely, Master Module, FEA Module, Optimization Module, and Data Retrieval Module.

DDiissppllaayy OOuuttppuutt

IInnppuutt ((GGUUII))

DDaattaa BBaassee

DD RR SS

GG A A

OOppeer r aattoor r ss

MMaasstteer r MMoodduullee

FFEE A A

FFii ttnneessss EEvvaalluuaatt iioonn

GG A A

CCoonnttr r ooll lleer r

Fig. 4 Software (BDS) Architecture

6.1 Master ModuleThe entire bridge design system is implemented in an object-oriented fashion in the Master Module. This module controls the design

process. The master module has a set of Graphical User Interface (GUI) through which the user interacts with the software both togive input data as well as to review output. All the user-defined inputs and controls are given through the GUI. The master module is

responsible for the finite element modeling, analysis and conventional design of steel truss girder railway bridge superstructure. Themaster module interacts with the optimization module for the optimal design of steel truss girder superstructure deck. The mastermodule shares data as well as methods with the optimization module to make the software more efficient with minimum code size.

During the design process, the master module documents the whole design in to a file as normally presented in professional designdocument. The design document thus generated can be used for manual design crosscheck as well as detailing of the steel truss

girder superstructure.

6.2 FEA ModuleThe FEA module models the three dimensional truss girder superstructure using beam elements (for main members) as well as trusselements (for bracings). This modeling is automated with only span of truss girder, number of panels and type of truss configuration

as input. This module interfaces with PASSFEM program to analyze the truss girder for various load combinations, consideringsecondary bending. The analysis result is documented in to an output file so that the governing design forces and secondary moments

of truss girder members can used for the conventional design as well as optimal design of steel truss girder superstructure.


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6.3 Optimization ModuleThe optimization module uses Genetic Algorithms (GA) as the optimization algorithm. This module consists of mainly three partsnamely, GA controller, GA operators, and fitness evaluator as shown in Fig 4. The function of GA controller is to guide the GA

process with the implicit knowledge it possesses. The implicit knowledge is used to fix up variable bounds as well as GA control parameters. For example if the user gives some infeasible bounds for the design variables and inappropriate values for GA control

parameters, the search process will be affected and it leads to uneconomical and even infeasible solutions. Thus the main function ofthe GA controller is the error handling based on the implicit knowledge. GA operations are the core part of the GA process. The

working of genetic algorithms is shown in Fig. 5. The following are the steps:1. The initial population is generated.2. The fitness is evaluated using the fitness evaluator.

3. If the population has not converged, GA operations such as reproduction, crossover, and mutation are carried out.4. The steps 2 and 3 are repeated till convergence occurs.

For evaluating the fitness, the fitness evaluator uses the functions in the design procedure in the master module. The GA process

interacts with master module in each and every generation. Once the convergence occurs, the optimal values of the design variablesare supplied to master module, which in turn performs conventional design check using the output data, for documentation purpose.

Best

Solution

FitnessEvaluation

No

Initial Population

Yes

Reproduction MutationCross-over

GA Operators

ConvergenceDecode

Ranking

Fig. 5 Working of Genetic Algorithms Fig. 6 Object-Oriented Implementation of Genetic Algorithms

The optimization module is implemented entirely in object oriented environment with the GA process implemented using the

concepts of object oriented method. The core GA attributes and methods, which are independent of the problem domain, arecompletely encapsulated and the access to the GA process from outside is given only through a well-defined interface. The features

available in object-oriented models such as ‘aggregation’ make the interface implementation easier. Fig. 6 shows the schematicdiagram, representing the object-oriented implementation of GA process. Object-oriented implementation of GAs as methods in a

class by abstracting the problem-independent genetic operations part and the problem-dependent function evaluation part has enabledeffective encapsulation, inheritance and better memory management.

6.4 Data Retrieval System (DRS)The static data containing the code provisions such as the allowable stresses and standard as a steel section properties are stored in a

relational database. The DRS is an object-oriented layer over relational database which enables the system to access data during runtime. The Open Data Base Connectivity (ODBC) is used as the methodology. The main component of DRS is a set of ODBC derivedclasses. These classes are connected to database, and to another intermediate set of classes, which are connected to the master

module. The schematic view of the DRS is shown in Fig. 7.

Database

(MS Access)

IRSCodeDatabase

HotRolledSteelSections

ODBC

Derived

Classes

(from MFC)

Intermediate Classes

(Contains retrieving

functions)

Data Retrieval System (DRS)

Master

Module

Fig. 7 Data Retrieval System (DRS)

Four specifications have been included in the database under design standards: three railway codes (Steel Bridge Code, 1977;

Bridge Rules, 1964; and Bridge Substructures and Foundation Code, 1985) and a part of wind code (IS: 875 (Part 3), 1987). The

GeneticAlgorithms

int *PopSize;

float *RealValues;

float Fitness;

float CumFitness;

void GenerateInitialPop( );

void Decode( );void ObjectiveFunction( );

void Reproduce( );void Crossover( );

void Mutation( );

void Replace( );

void CheckConvergence( );

Master Module

FEA Module

(PASSFEM)

Interface

Interface


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static data repository (MS Access) stores the basic steel section properties given in SP: 6(1) (1964). The ‘IRSCodeData’ database inMS Access consists of totally 22 tables in which 13 tables are for the standard loading of modified broad gauge which are given inIRS Bridge Rules and the remaining tables are for the allowable compressive stresses governed by slenderness ratio, allowable

tensile stresses governed by fatigue and allowable bending compressive stresses as governed by lateral buckling.

7. Optimal Design Formulations of Steel Truss Girder SuperstructuresThe design variables, objective function, a set of constraints that define the limits on the design problem have to be identified.

7.1 Genetic Design ModelThe genetic algorithms are implemented in object-oriented programming language (C++) because it has rich string manipulation

functions, and offers powerful techniques for data structuring, which is essential for effective representation and efficient processingof information. The inputs required to the program are of two types: the parameters which control the GA search, such as populationsize, crossover and mutation probabilities, convergence parameter etc., and problem specific data. The genetic parameters are given

through a dialog box, whereas, problem specific data is read from files. A detailed description of genetic modeling of steel trussgirder railway bridges is given below.

The cost of the through type truss girder superstructure is to be optimized. Typically, cost is a function of total structural weight.Cost of connection is the other major factor that affects the cost of the structure. In the optimal design process, the topology, shape

and cross-sectional parameters of the members of the superstructure are arrived at appropriately, so that the weight of the structure isminimal. All the members of the superstructure are built-up sections using plates of different thicknesses. Different topologies are

generally obtained by varying the number of panels and the one type of web pattern to be used in all panels, from among different patterns (fig. 8).

Genetic modeling consists of defining an individual string to represent a design and a corresponding coding, decoding scheme.The members of the structure are classified as stringers, cross girders, top chords, bottom chords, diagonals, verticals, portal girders,

sway girders, bottom chord bracings and top chord bracings. It is not economical to have the same cross-sectional dimensions for allthe members of the class, particularly in the case of the top chord, bottom chord and web members. An adaptive member grouping

strategy is implemented that groups members automatically during the evolution process. The members of top chords, bottom chords,diagonals and verticals having different dimensions are chosen from the possible alternate sections that are normally used in theconventional design process. A number of different cross-sectional shapes are possible for each member. But the shape of the cross

section of the member within each class is kept the same for all members belonging to same type, although dimensions may bechanged. From aesthetical considerations and joint fabrications considerations, usually overall dimensions of the chord member are

maintained constant over the entire length and only thickness of the elements are changed. In addition to the chord members and webmembers group, there is one group each for end diagonal, stringers, cross girders, portal girder, sway girder, bottom lateral bracingsand top lateral bracings. Thus different members to be designed are identified. Similarly, different web configurations are identified

to arrive at optimum truss girder during the evolution process.Member grouping is treated as a two-step process. Initial member grouping is done with same section for all top chord members

and so also for web members and bottom chord members. The web configuration is also fixed and a preliminary finite element

analysis is carried out and then fitness is evaluated. Next generation onwards, a regrouping which has different sections for differentmembers is done dynamically and the genetic design evolves over a number of generations. Every generation, the genetic operationstake place and the evolution process continues till convergence occurs. Generally, a value of 70 to 90 % of the population reaching

the best fitness is assumed as convergence. Convergence parameter is taken as the percentage of the population in the generationhaving the design variables corresponding to the best fit function.

(a) Warren Truss (b) Pratt Truss (c) Pratt Truss with

Subdivided Struts

(d) Pratt Truss with Subdivided Ties (e) Warren Truss with Subdivided Members (f) K Truss

Fig. 8 Web Configurations for Truss Girder Bridges

7.2 Design VariablesThe success of an optimization problem depends on the identification of design variables properly. The design variables contribute

mainly to the total cost. The design variables should be selected such that they completely represent the design space and contributeto the objective function. The design variables should be independent of each other. The sensitivity of the selected design variables

also plays an important role in the final convergence to the optimal solution. As the number of design variables increases, and as theequations become nonlinear the complexity of the optimization problem also increases. Most mathematical optimization techniques

become cumbersome and even fail due to their inability to handle complex design space with a large number of mixed type design


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variables.. GA can handle discrete, integer, real and boolean design variable with equal ease. The sizing design variables consideredin this study are either cross-sectional dimensions using continuous variable for lateral dimensions and discrete design variables forthickness. The standard cross section is described in terms of integer pointing to a row in a table of available cross sections. A

boolean variable is used to specify parallel chord system and non parallel chord system. Table 1 shows the design variablescorresponding to the steel truss girder superstructure optimization problem.

Table 1 Linking of Design Variables (DV) and the Physical Meaning

Optimization Physical Meaning DV Type in GA Note

Truss System Truss Configuration BooleanIndicates whether Parallel Chord

or Non Parallel Chord System

SizingStandard Cross Section,

Thickness selectionInteger Search through a given table

ShapePlate Width,

Joint CoordinatesReal

Varies between upper and lower

bounds

A typical truss configuration used in Indian Railway (Fig. 9) is taken for illustrating member grouping, selection of crosssections, identification of continuous and discontinuous variables, choice of different bracing patterns and their influence on the

string length used to represent the design variables.

8 Panels

h

T1 T2 T3

B1 B2 B3 B4

D1 D2 D3 V1 V2 V3 V4

Fig. 9 Typical Truss Configuration

There are three different top chord member, four bottom chord member, three diagonals and four vertical members, takingadvantage of symmetry. The shape of the member is kept the same for members belonging to the same type. In addition to this, there

is one group each for each type of end diagonal, stringers, cross girders, portal girder, sway girder, bottom lateral bracings and toplateral bracings. Thus 21 different member groups are identified for this particular problem. Number of different possible shapes

exists for each type of members (Fig. 10). A discrete variable for representing a cross section of each member in the truss girder suchas top chords, bottom chords, end diagonals, diagonals, verticals, portal girders, sway girders, top lateral bracings and bottom lateral bracings is used in the genetic model. Thus up to 9 discrete variables are needed to represent the shape of the cross section.

31 24 5

6

7 8 910

11 12b

W

t1

t2

W

D

b

t1

t2D

b

t 1 t2

W

(b) Bottom Chords – 4, 5

(c) Diagonals and Verticals – 6, 7, 8 (d) Portal Girders and

Sway Girders - 9

W

D

b

t1

t2 D

b

t1

t2

W

D

b

t1

t2

(e) Lateral Bracings- 10, 11, 12

b

W

D

t1

t3

t2

W

D

b

t1

t2

(a) Top Chords and End Diagonals – 1, 2, 3

b1

W

D t 1

t 3

t2

b2

Fig. 10 Shapes Considered for Different Welded Members with ID Numbers

The thicknesses of plates (10mm, 12mm, 14mm, 16mm, 18mm, 20mm, 22mm, 25mm, 28mm, 32mm, 36mm, 40mm, etc.) to beused are stored sequentially in a discrete set database. From the fabrication and aesthetic point of view, it is required to keep a few parameters constant throughout the bridge. The possible web bracings are also numbered to be represented as discrete variables. For

instance, width of all the main members (top and bottom chords) is generally kept constant so as to be able to connect them properlyto gusset plate. Similarly depth of all top chord members is kept same so also the depth of all bottom chord members. Depth of

diagonals and verticals then depends on the width of bottom and top chord members and thicknesses of gusset plate used. The detailsof an individual string are given in Table 2. For continuous design variable values sixty four (2 6) possibilities are considered, whereas

for discrete design variable values eight (23) possibilities are considered. Coding scheme is illustrated in Fig. 11.


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Table 2 Composition of Individual Strings

No of Variables Total Bit SizeMember Groups

Continuous Discrete Continuous Discrete

No ofGroups

Sub-stringLength

Web Configuration 1 3

Shape of Sections 1 9 27

Members Width (W) 1 6

Top Chords D

b & ti

1

1 3

6

6 9 3

6

45

Bottom Chords D

b & ti

1

1 2

6

6 6 4

6

48

End Diagonals b & ti 1 3 6 9 1 15

Diagonals D, b & ti 2 2 12 6 3 54

Verticals b & ti 1 2 6 6 4 48

Portal Girder d, ti 1 2 6 3 1 9

Sway Girder d, ti 1 2 6 3 1 9

Top Lateral Bracings 1 3 1 3

Bottom Lateral Bracings 1 3 1 3

Total string length 282

Note: 6 bits for continuous variable and 3 bits for discrete variables

1 2 3 p

1 2 3 n 1 2 3 m

p-1

n-1 m-1

1 – p Population size

1 – n Continuous variables 1 – m Discrete variables

1 2 3 4 5 6

Substringlength (6)

1 2 3

Substringlength(3)

Fig. 11 Illustration of Coding Scheme

7.3 Objective FunctionIn the present study, the aim is to minimize the cost expressed as the objective function. The total cost of superstructure comprises

material costs and fabrication costs. Joints costs depend upon the number of joints, the number of members joining at the node andthe forces to be transferred at the joint. In the present study, various web bracing configuration is considered in the genetic evolution

process. The constrained form of the optimization problem can be formulated as

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ∑ ∑+×⎟⎟

⎠

⎞⎜⎜⎝

⎛ ∑ ××=

= ==

j mjmn

j

n

r r st

N

iii C C l A MinimizeC

1 11

ρ (6)

Subject to the following constraints:

• Material strength, buckling strength as well as fatigue strength limit.• Deflection limit.Where C = objective function; Ai, l i and ρ = area, length and weight density of the ith member, respectively; C st = cost of steel perkN; nmj = number of members meeting at the nodes and Cr = cost of joining each member depending upon its force.The objective of the optimum design problem is to minimize the total cost of the superstructure, which is represented by Eq. 6.

7.4 ConstraintsAn optimal solution is the best solution among a set of feasible solutions, which satisfy all the constraints imposed upon them. InGenetic Algorithm based optimal design process, the constraints can be implicit constraints or explicit constraints.

• The implicit constraints on the design variables can be represented by choosing appropriate values for the lower and upper bounds on the design variables.

• Explicit Constraints are basically functional constraints, which are evaluated based on the design variables. In the present study,explicit constraints are imposed on material strength, buckling strength as well as fatigue strength limit and deflection limit.Further, the unsupported projection of any plate, measured from its edge to the line of rivets, bolts or weld connecting the plate

to other parts of the section is limited to 16 t for steel conforming to IS : 226 and IS : 2062 and 14 t for steel conforming to IS :961. The IRS Steel Bridge Code specifies maximum slenderness ratio limits for compression members, tension members and bracing systems. Theses limits become constraints in the respective member design.

The genetic algorithms can be used for optimizing unconstrained objective functions. In this implementation of GA the constrained problem is converted into unconstrained problem, using penalty function, to obtain an augmented objective function and fitness

values, as given below:

( )caug PC f +×= 1 (7)


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8

( )∑ ×+×+×= iimplicit d deflectionsstressc PC PC PC P (8)Where f aug = Augmented Objective Function; C = Objective function given by Eq. 6; P c = Total penalized constraints violations;Cstress, Cdeflection and Cimplicit = stress, defection and implicit constraint violations respectively; and P s, Pd and Pi = penalty factors for

stress constraints, deflection constraints and implicit constraints, respectively.To evaluate the fitness, Fi, of a solution, the value of the augmented objective function has to be subtracted from a large number.

Usually this large number is taken as the sum of the maximum and minimum objective function values in the population (Eq. 9).

iaugi f f f F

,minmax −+=

(9)Where f max, f min = maximum, minimum value of augmented objective function in a generation.

The optimal design of the truss bridge in Fig. 9 is carried out using the Genetic Algorithms module assuming the same trussconfiguration as in standard design and optimizing only member dimension (BDS Optimal Design I). The Genetic parameters used

for the optimal design of steel truss girder superstructure had a population size = 300, probability of crossover = 0.8, probability ofmutation = 0.002, penalty function coefficient = 1. The tournament selection process and single point crossover have been used forarriving at the optimal design. The optimal design output of the software for a steel truss girder superstructure having a span length of

47.24 m is compared with the standardized design of Research Designs and Standards Organization (RDSO). Sections obtained based on optimum design are compared with RDSO

sections in Table 3. The total weight of steel iscompared and shown in Fig. 12. It is seen thatoptimally designed truss bridge weighs less by around

25 % (Fig. 12, BDS Optimal Design I).In addition to 25 % reduction in steel costs,

possibly an improvement of another 5 to 10 % costsmay result if fabrication costs of members, joints costsand changing web bracing configurations could be

allowed in the genetic model. However, in order tovalidate the optimal design obtained from the BDS, the

same cross section shape for members and webconfiguration adopted in the RDSO design are

maintained for the genetic model. The same truss bridge is optimized by allowing different cross section

shape for members, web configuration and number of panels. Around 33 % reduction (Fig. 12, BDS Optimal

Design II) in weight is observed when all the designvariables are chosen based on optimization.

Fig. 12 Weight Comparison of Optimal Design with RDSO Design

Table 3 Comparison of Sections obtained based on Optimum Design and RDSO SectionsDescription ID W, mm d, mm b, mm t1, mm t2, mm t3, mm

BDS 9 720 360 20 10Stringers

RDSO 9 750 360 20 10

BDS 9 960 430 22 10Cross

Girders RDSO 9 935 500 20 10

BDS 1 610 520 150 18 16 18T1

RDSO 1 642 542 150 20 16 20

BDS 1 610 520 150 18 16 18T2

RDSO 1 642 542 150 20 16 20

BDS 1 610 520 150 18 16 18

Top Chords

T3 RDSO 1 642 542 150 20 16 20

BDS 3 578 490 150 20 20

B1 RDSO 3 610 542 150 25 20

BDS 3 578 490 150 20 20B2

RDSO 3 610 542 150 25 20

BDS 3 578 490 150 20 20B3

RDSO 3 610 542 150 25 20

BDS 3 578 490 150 20 20

Bottom

Chords

B4 RDSO 3 610 542 150 25 20

BDS 3 578 400 100 16 16D1

RDSO 3 6102 # ISMC 400x1002 # Inner Plate 150x12

BDS 3 578 400 100 16 16D2

RDSO 3 6102 # ISMC 400x1002 # Inner Plate 150x12

Diagonals

D3 BDS 3 578 400 100 16 16

Comparison of Weight of Steel

0

100

200

300

400

500

600

700

RDSO Design BDS

Optimal Design I

BDS

Opt imal Design II

W e i g h t ( k N )


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9

RDSO 3 610

2 # ISMC 400x1002 # Inner Plate 150x12

BDS 9 554 200 12 10V1

RDSO 9 586 200 12 10

BDS 9 554 200 12 10V2

RDSO 9 586 200 12 10

BDS 9 554 200 12 10V3 RDSO 9 586 200 12 10

BDS 9 554 200 12 10

Verticals

V4 RDSO 9 586 200 12 10

BDS 9 500 200 10 10Portal Girders

RDSO 9 522 200 10 10

BDS 9 500 200 10 10Sway Girders

RDSO 2 # ISST 150x75x8

BDS ISA 150x150x10Top Lateral Bracings

RDSO ISST 200x165x8

BDS 2 # ISA 100x100X8Bottom Lateral Bracings

RDSO 2 # ISA 100x100x8

BDS 435 (25 % lesser than RDSO Design)Total Weight (kN)

RDSO 583

8. Summary and ConclusionsThe paper demonstrates application of GAs for optimum design of truss type Railway bridges. The main thrust of this paper is to

illustrate typical application of GAs to practical design of structural systems such as steel truss girder railway bridge superstructure inan object oriented implementation of design and optimization codes. In this work binary strings for coding of truss girder designvariables, ‘fitness’ as a ranking measure of the adaptability to the environment, selection criteria and GA operators such as crossover

and mutation are used to improve the fitness of the population, so that over the generations the genetic algorithm progresses towards better design variables and at the end converges to the optimum value. A sample problem solved using the software shows

considerably more economical design can be obtained using GA compared to a standard design. There is scope for wider applicationsof GAs based methodologies in optimal design of practical structures considering system configuration and member sizes as design

variables.

9. AcknowledgementThis work is supported by DST and RDSO, Ministry of Indian Railways. The presenting author is grateful to University ofCambridge Department of Engineering, Jesus College Cambridge, and the Gates Cambridge Trust for kindly granting the funding to

attend the congress. These sources of support are gratefully acknowledged.

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