J. Software Engineering & Applications, 2011, 4, 571-578
doi:10.4236/jsea.2011.410066 Published Online October 2011 (http://www.SciRP.org/journal/jsea)
571
Genetic Algorithms-Based Optimization of Cable
Stayed Bridges
Venkat Lute1, Akhil Upadhyay2, Krishna Kumar Singh3
1Civil Engineering Department, Gayatri Vidhya Parishad College of Engineering, Visakhapatnam, India; 2Civil Engineering Depart-
ment, Indian Institute of Technology Roorkee, Roorkee, India; 3Civil Engineering Department, Indian Institute of Technology Roor-
kee, Roorkee, India.
Received July 20th, 2011; revised August 18th, 2011; accepted September 15th, 2011.
ABSTRACT
Optimum design of cable stayed bridges depends on number of parameters. Design of Cable stayed bridge satisfying all
practical constraints is challenging to the designers. Considering the huge number of design variables and practical
constraints, Genetic Algorithms (GA) is most suitable for optimizing the cable stayed bridge. In the present work the
optimum design is carried out by taking total material cost of bridge as objective function. During problem formulation
most of the practical design variables and constraints are considered. Using genetic algorithms some parametric stud-
ies such as effect of geometric nonlinearity, effect of grouping of cables, effect of practical site constraints on tower
height and side span, effect of bridge material, effect of cable layout, effect of extra-dosed bridges on optimum relative
cost have been presented. Data base is prepared for new designers to estimate the relative cost of bridge.
Keywords: Cable Stayed Bridge, Genetic Algorithms, Stiffness Analysis
1. Introduction
The modern cable-stayed bridge consists of a superstruc-
ture of steel or reinforced concrete members supported at
one or more points by cables, extending from one or
more towers. The renewal of the cable-stayed system in
modern bridge engineering was due to the tendency of
bridge engineers to obtain optimum structural perform-
ance from material which was in short supply during the
post-war years [1]. Due to their aesthetic appearance,
efficient utilization of structural materials and other nota-
ble advantages, cable-stayed bridges have gained popu-
larity in recent decades. This fact is due to the relatively
small size of the substructures required and due to the
advent of efficient construction techniques apart from the
rapid progress in the analysis and design of this type of
bridge. Dimensioning of particular bridge from economic
consideration, meeting the safety and serviceability re-
quirements is complicated due to wide possible range.
Cable stayed bridges are statically indeterminate stru-
ctures and their structural behavior is largely affected by
the cable arrangement, and stiffness distribution in the
cable, deck, and pylons. The analysis of cable stayed bri-
dges is discussed by [2,3 ], but the analysis is limited to a
fixed geometry because it is based on impirical formulae.
References [4,5] have carried out parametric studies re-
garding the effect of cable, girder, and pylon stiffness on
bridge behavior considering only linear analysis. Thus, in
the present work rational approach for the analysis, in-
cluding nonlinearities of cable stayed bridge is consid-
ered. Since GA requires very efficient and reasonable
accurate analysis, the present stiffness based approach is
most suitable for GA optimization. References [6,7] have
investigated the cable anchorage position in girder,
side-to-main span ratio and number of cables individually.
However, the studies of effect of individual parameter on
relative optimum cost of bridge do not carry much sig-
nificance. Thus in the present work the optimization is
carried out by considering more design parameters as
design variables simultaneously. Reference [8] used GA
for space truss optimization and [9] used the GA for the
optimum design of fiber composite stiffened panels to
study the robustness of GA in handling mixed types of
variables. Reference [10] studied the application of ge-
netic algorithms on optimum design of bridge decks. In
optimizing cable stayed bridges, the cross sectional area
of stiffening girder, tower and cable are crucial and are
decided based on the strength and stability criteria. Ref-
erence [11,12] carried out the optimization of cable stayed

Genetic Algorithms-Based Optimization of Cable Stayed Bridges
572
bridges for cross sectional area of members. The entropy
based optimization approach adopted by them will pose
limitation in considering many design variables simulta-
neously. In the present paper all the possible design pa-
rameters which affect the optimum cost of bridge, sig-
nificantly, are considered as design variables and all
types of constraints, strength, stability and serviceability
are incorporated in the optimization routine.
1.1. Analysis
Stiffness method is used for analysis of radiating type of
cable stayed bridge using MATLAB. It is assumed that
the girder is passing between tower legs and supported on
rollers. The cables are fixed to the tower top. Side span is
resting on hinge support. For the modeling purposes the
cables are considered as no-tension truss elements, girder
and tower as beam element. At each node of the beam
element three degree of freedom, horizontal, vertical and
rotation are considered. For cables only, two degrees of
freedoms are considered. The details of bridge geometry
are shown in Figure 1. The input parameters of the bridge
model are given in Table 1.
D
f
H
t
L
s
L
m
L
s
BL
L
us
Bridge geometry
B
3
B
2
B
1
T
g
T
g
Girder section
Tower section
B
4
B
5
T
t
Figure 1. Details of idealized bridge.
Table 1. Input for bridge model.
Input for bridge model PropertiesQuantity Remark
Tower Et 2.1E+11 N/m2
Girder Eg 2.1E+11 N/m2
Ec 1.65E+11 N/m2
Cable Ac 0.01 to 0.10 mvariable
Dead load DL 50 k N/m
Live load LL 40 k N/m
Ratio of central Unsupported length to main spa n, Lus/Lm 0.05 to 0.4 Variable
Ratio of side span to main span, Ls/Lm 0.35 to 0.55 Variable
Ratio of tower height to main spa n , Ht/Lm 0.2 to 0.4 Variable
Number of cables 8 to 68 Variable

Genetic Algorithms-Based Optimization of Cable Stayed Bridges573
Problem Formulat ion
In the present work, the stiffness approach is used to per-
form nonlinear static analysis of two dimensional cable
stayed bridge. Two-dimensional static analysis will be
adequate for all practical purposes in optimization of ca-
ble stayed bridges. The stiffness method was particularly
adopted to handle various types of structural elements
such as cables, and beam-column elements and it can be
conveniently adopted for computer use. While modeling
the connection between girder and tower, the connection
node are separately considered and boundary considered
are applied independently. In other words, the junction
node is not merged. Full geometric nonlinearity, arising
from cable sag, beam-column effect and large deformation
has been taken in to account while analyzing the struc-
ture.
1.2. Load Cases
Two main load cases were considered for the final struc-
ture, the action of dead load and the action of live load.
The action of live load consists of four patterns: over
entire deck, over outer spans only, over main span only
and over half bridge only. During the optimization the
analysis routine analyses the structure for maximum re-
sponse under any live load pattern at any section. Dead
load and live load are accounted for as uniform loads in
the stiffening girder as 50 kN/m and 40 kN/m respec-
tively. The dead load includes, concrete slab over stiff-
ening girder and the self weight of structural members.
2. GA Based Formulations
GA deals with population that is collection of candidate
solution. A population is a collection of N individuals.
An important feature of a population, especially in the
early generation of its evolution, is its genetic diversity.
The too small population size may lead to scarcity of
genetic diversity. It may result in a population dominated
by almost equal chromosomes and then, after decoding
the genes and evaluating the objective function it may
converge quickly but may lead to local optimum. At the
other extreme, in too large populations, the overabun-
dance of genetic diversity can lead to clustering of indi-
viduals around different local optima. But the mating of
individuals belonging to different clusters can produce
children lacking the good genetic part of either of the
parents. In addition, the manipulation of large popula-
tions may be excessively expensive in terms of computer
time. Thus proper selection of population size is ex-
tremely important. Thus over few trials, the population
size 60 has been adopted.
The GA based optimization basically d epends on three
important aspects viz.: 1) coding and decoding of design
variables; 2) evaluation of fitness of each solution string;
and 3) application of genetic parameters (selection, cross-
over and mutation) to generate the next generation of
solution strings.
2.1. Design Variables
The formulation of an optimization problem begins with
identifying the underlying design variables. Generally,
the parameters such as tower height, unsupported length
of main span, side span are expressed as percentage of
main span length (Krishna et al., 1985; Hegab, 1988). In
the present task the selected design variables are:
 Cable areas (Ac),
 Girder dep t h (B3),
 Girder bottom width (B 2),
 Girder top width (B1),
 Girder we b thicknes s ( T g),
 Tower width (B 4),
 Tower dept h (B5),
 Tower web thickness ( T t),
 Tower height-to-main span ratio (Ht/Lm),
 Side-to-main span ratio (Ls/Lm),
 Central unsupported length-to-main span ratio (Lus/
Lm),
 Number of cables (Nc) and Bridge type (harp or radi-
ating).
2.2. Coding and Decoding of Variables
An essential characteristic of genetic algorithm is the
coding of variables that describe the problem. The vari-
ables in the design space are converted into genetic space
by transforming the variables in binary form. The most
common coding method is to transform the variables to a
binary string of specific length of the chromosome. For a
specific problem that depends on more than one variable,
a multivariable coding is constructed by simply jo ining as
many single variables coding as the number of variables
of the prob lem. The sequ ence of vari ables and the co ding
for the present problem is illustrated in Figure 2.
2.3. Objective Function
In structural design, the dominant objective will be mini-
mizing structural cost. There can be multi objective func-
tions in one problem, but generally it is avoided by choos-
ing the most important objective as the objective func-
0 0 000 00
L
s
/L
m
H
t
/L
m
L
us
/L
m
B
1 B
2
0 000 0 0 0
A
c1
B
3
T
gf
B
4
B
5
T
tf
A
c2
A
c3
A
c4
0
N
c
0
Bridge type
Figure 2. Variable’s seque nc e and coding.

Genetic Algorithms-Based Optimization of Cable Stayed Bridges
574
tion and the other objective functions are included as
constraints by restricting their values with in a certain
range. The objective function for the present work is ex-
pressed in terms of relative cost of bridge material, given
by Equation (1). Generally, structural design is required
to conform to number of inequality constraints related to
stresses, deflection, dimensional relationships, and other
codal requirements. These are conveniently handled in
genetic algorithms by penalty approach. In this approach
a candidate solution is penalized for constraints violation
while determining the future use of that solution. The
level of penalty depends on the criticality of the role of
constraints in objective function. Since design of cable
stayed bridge is constrained problem, the objective func-
tion should be transformed in to unconstrained problem
by introducing an exterior penalty fun c tion.
The objective function is expressed in terms of design
variables and constraints to be satisfied. Mathematically,
for the present problem, for a set of “x” design variables
the objective function is expressed in terms of cost of
cable, cost of structural steel of tower and girder.
Cost ratio is defined as ratio of cost of cable material
per unit volume to the cost of Girder and tower material
(steel or concrete) per unit volume. If steel material is
used for girder and tower the cost ratio is taken as 2 and
if concrete material is used for girder and tower the cost
ratio is taken as 10.

c
N
1
Cost ratio2
ii ggtt
i
f
xLALA


HA (1)
where Li = length of ith cable, Ai = Area of ith cable; Lg =
girder length; Ag = girder area; Ht = tower height; At =
tower area. Nc = Number of cables.
The modified objective function with penalty factors is
expressed in Equation (2)
 
1
1
pi
a
ii
g
xfxK g






 




1

(2)
If 1
i
a
i
g
g the objective function will not be penal-
ized, i.e., 10
i
a
i
g
g .
where K is penalty parameter, ia
g
is allowable value of
constraint gi, p is number of constraints. The penalty pa-
rameter is obtained by trial. When the fitness is to be
maximized the modified objective function reach to a
minimum value.
2.4. Constraints
The constraints represent some functional relationships
among the design variables and other design parameters
satisfying certain physical phenomenon and certain re-
source limitations. Some of these considerations require
that the design remain in static or dynamic equilibrium.
All practical types of constraints, related to stress, stabil-
ity and strength are incorporated in the present work. The
constraints have been expressed in normalized form in
Equation (3)
constraint violation1;
i
ia
i


 (3)
2.5. Fitness Evaluation
The evaluation of the fitness is problem dependent as
well. For every individual in the population, a level of
fitness must be assigned to it. The fitness of an individual
is an indicator of how well an individual is suited to its
current environment. Fitness is established by means of a
function. Developing this function can be very simple or
may be very complex involving a simulation.
The population size is generally selected as approxi-
mately equal to chromosome length. As al ready di scussed,
the binary coding is more general form of coding in GA
and it is easy to implement. In order to retain some num-
ber of best individuals at each generation the elitist selec-
tion is chosen in GA operator. To reduce positional bias
and end-point effect two-point crossover has been chosen
and to achieve quick convergence, with few trails the
mutation rate was selected. In the optimization of cable
stayed bridge the GA parameters considered are listed in
Table 2. The process of optimization for the design of
cable stayed bridge is illustrated with the help of flow-
chart give n in Figure 3.
2.6. Pseudo Code for GA Implementation
% declare size of population and variables
nov = number of variables
sop = size of population
nob[1: nov]= number of bits for each variable
Step 1: Generate Initial Population
for i = 1: sop
Table 2. GA input parameters.
Parameter Coding Selection Crossover MutationPopulation size
Method binary Elite-count2-point; 0.7%0.001 60

Genetic Algorithms-Based Optimization of Cable Stayed Bridges575
Input Data
Generate initial population
Apply Genetic operators,
(Reprodu cti on , crosso v er, and mutation)
Decoding of Design variables
Evaluate constraints,
Objecti ve funct ion and
penalized objective function
No
GEN= GEN+ 1
Convergence check
Yes
Print
Optimum
solution
Figure 3. Flow chart for GA optimization.
for r = 1: sum(nob)
binpop(i, r) = generate random
number in binary form;
end
end
while (current generation <= maximum generation)
% For each generation do following
{
Step 2: Binary assignment of variables in
chromosomes
Step 3: Fitness Calculations For each chromo-
some
Get structural responses
Compare with permissible values
Penalize the constraint for violation
Modified objective = objective with penalty
Find fitness using modified objective function
Fitness factor = fitness of each chromosome/
average fitness;
Step 4: find index for fittest chromosome
Step 5: Applying Genetic Operators
Elite-count Selection
2 point Crossover
Mutation with appropriate rate
Termination criteria
}
3. Parametric Study
In order to validate the optimization studies parametric
studies are carried out. The various parameters such as
effect of bridge material, site restriction on tower height,
side span, cable layout, effect of cable grouping, geomet-
rical nonlinearity and extra-dosed bridges on optimum
relative cost has been studied.
3.1. Effect of Penalty Parameter on Convergence
Penalty parameter (K) in Equation (3) is the important
parameter which imposes the penalty on objective func-
tion based on the constraint violation. In the present
problem the penalty parameter is varied from 1 to 9 and
the optimization process is carried out. It was observed
that higher the penalty parameter longer time taken for
convergen ce as shown in Figure 4. For the present prob-
lem by keeping the maximum generations fixed, for
higher penalty factor the objective function is more. De-
pending upon the sensitivity of constraint the constraint
violation is penalized. Unnecessarily increasing the pen-
alty constant will take solution far away from boundary
leading to late convergence. Thus the K = 1 is appropriate
for the present problem.
3.2. Effect of Bridge Material on Optimum
Relative Cost
The material considered for the present study of cable
stayed bridge, are as follows:
1) Tower and deck/girder as concrete and cable high
tensile steel;
2) Tower and deck/girder as steel and cables as high
tensile steel.
For the calculation of cost, the cost ratio of cable steel
to structural steel is taken as 2. While studying the con-
crete bridge the cost ratio of cable to concrete is taken as
6, 8 and 10. The optimum relative cost for both cases is
Figure 4. Effect of penalty parameters on objective function.

Genetic Algorithms-Based Optimization of Cable Stayed Bridges
576
compared in Figure 5. It is observed that for 500 m
bridge even with cost ratio of 10 the reduction in steel
cost is 45%. It indicates that for long span bridges it is
advisable to use steel than concrete.
3.3. Effect of Geometric Nonlinearity on Relative
Bridge Cost
The results of design optimization using linear and non-
linear analysis are shown in Figure 6. In this figure, the
optimum relative cost of four bridges with bridge lengths
of 60 m to 500 m are presented using both linear (first-
order) and nonlinear (second-order) analysis results. It
can be seen that there is an increase in optimum cost if
nonlinear analysis results are used for bridge lengths
above 250 m. This increase is proportionate to the bridge
length. For the bridge length of 150 m, there is an in-
crease of 13% in the optimum cost due to consideration
of nonlinear effects. For higher bridge lengths, the in-
crease can be much more.
Figure 5. Effect of bridge material.
Figure 6. Effect of geometrical nonlinearity on optimum re-
lative cost.
3.4. Effect of Grouping of Cables
GA-based design optimization on two types of bridges: 1)
bridge with the same cross-section for all the cables; 2)
bridge with four different cross-sections for cables in four
zones. The effect on optimum relative cost of cable mate-
rial has been studied. It is observed from the Figure 7
that from 120 m bridge length the grouping of cable will
reduce the optimum cost by 5%. Instead of adopting the
same area for all the cables, it is advantageous to give
larger areas to outer most cables with respect to pylon
and lesser areas to interior cables. For this purpose, the
cables in the bridge are divided into four zones. It is ob-
served that reduction in the cost is directly affected by the
reduction in the cable volume.
3.5. Effect of Practical Site Constraints: Tower
Height
Many a times the optimum ratio for tower height-to-main
span ratio may have to be altered based on site conditions.
In such situation, for given tower height-to-main span
ratio the optimization is carried out to get the optimum
cost. In the present work the practical tower height-to-
main span ratios 0.1, 0.2, 0.3, and 0.4 are considered. For
all these ratios the optimization stud y has been conducted.
It is observed from Figure 8 that as bridge length is in-
creasing the optimum tower height-to-main span ratio has
shifted from 0.2 to 0.4.
3.6. Effect of Optimum Side-to-Main Span Ratio
on Relative Cost
In some situations the designer is restricted to use par-
ticular side-to-main span ratio based on site conditions.
For that purpose this study is conducted to facilitate the
designer to get optimum cost for the restricted side-to-
main span ratio. The ratios studied here are 0.2, 0.35, 0.4,
0.45, 0.5, 0.55, and 0.75. The relative optimum cost for
each bridge length has been studied. It has been observed
Figure 7. Effect of grouping of cables on optimum relative
cost.

Genetic Algorithms-Based Optimization of Cable Stayed Bridges577
Figure 8. Effect of tower height restriction on relative opti-
mum cost.
from Figure 9 that as bridge length increases the opti-
mum side-to-main span ratio decreases from 0.55 to 0.35.
This study reveals the fact that although the practical range
of side-to-main span ratio is 0.35 to 0.55 the exact ratio
for particular bridge length can thus be obtained.
3.7. Effect of Practical Site Constraints:
Cable Lay Out
Two types of cable layout are considered for the present
studies, one with radiating type and second with harp
type. Although it is a design variable, still for the purpose
detailed understanding the effect of cable layout this
study has been carried out. The optimum design results
are compared for two types of cable layout by restricting
the design variable to particular cable layout. The relative
optimum cost is plotted with respect to various bridge
lengths in Figure 10. Keeping all parameters same, only
relative optimum cost variation is compared. It is ob-
served that for up to 300 m bridge length the optimum
cost has no difference. Above 300 m bridge lengths the
radiating type of bridge cost around 5% to 12% less.
Hence the radial type is recommended from economy
point of view when the static case is considered.
3.8. Effect of Extra-Dosed Bridge
The extra-dosed bridges are designed for lesser tower
heights. For this study the tower height-to-main span ra-
tion has been restricted to 0.05 to 0.2. The study revealed
that the extra-dosed bridge found to be 3% economical up
to 250 m bridge length (Figure 11). But beyond the
bridge length of 300 m the extra-dosed bridges are cost-
lier around 12% as compared to ordinary cable stayed
bridges.
4. Summary and Conclusions
GA is robust tool for optimization. In the present work
stiffness based analysis approach is used for cable stayed
bridge. Considering all the practical constraints and most
crucial design variables the optimum design studies are
Figure 9. Optimum side-to-main span ratio.
Figure 10. Effect of cable layout on optimum relative cost.
Figure 11. Effect of extradosed bridges on optimum cost.

Genetic Algorithms-Based Optimization of Cable Stayed Bridges
578
carried out using genetic algorithms. The following con-
clusions are drawn from the present work:
 GA can handle more number of variables easily. The
developed GA program is more general to accommo-
date discrete and continuous variables
 Effect of bridge material study reveals that for 60 m
bridge length decrease in relative cost for steel bridge
is around 25%. As bridge length increases from 60 m
to 500 m there is increase in the reduction in steel
bridge cost up to 45% as compared to concrete bridge
cost.
 Due to inclusion of cable grouping concept in the
current optimization approach, reduction in the rela-
tive cost is observed. As bridge length increases from
60 m to 500 m the reduction in relative optimum cost
varies between 5% to 13%. It is expected more when
the bridge length increases beyond 500 m and in
situations where cable numbers are more.
 By incorporating the geometric nonlinearity in the
analysis of cable stayed bridge in optimization ap-
proach up to 200 m bridge length there is hardly any
difference in optimum relative cost. An increase in
optimum relative cost was observed for bridge length
ranging from 250 m to 500 m up to 13%.
 It is observed, by restricting the tower height, that the
optimum tower height-to-main span ratio is shifted
from 0.2 to 0.4 as bridge length increased from 60 m
to 500 m.
 The effect of optimum side-to-main span ratio on op-
timum relative cost study revealed that the optimum
side-to-main span ratio is changing from 0.55 to 0.35
as bridge length increases from 60 m to 500 m. The
side-to-main span ratio is shifted towards 0.35 fol-
lowing s-curve pattern.
 When the cable layout is restricted to radial type only,
up to 300 m bridge length the difference in optimum
relative cost is neglig ible. Beyond 300 m length , it has
been observed that radial-type cable layout can result
in reduction of the optimum cost by 5% - 12% from
the bridge with harp type cable layout.
The data base prepared for various practical bridge leng-
ths will be beneficial for designers to estimate the cost of
cable stayed bridges.
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