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CHAPTER 2

LITERATURE REVIEW

2.1 General

As the retractable roof stadium occupies a very huge area and consumes large amounts of

material for its construction, optimal design of such a structure is of utmost importance.

Hence, the literature pertaining to the status of the retractable roof structures together with

the developments that have happened in the field of structural optimization are taken-up

here. The earlier research on the retractable roof structures was mainly concentrated on the

geometric design of the roof structure, giving less or no significance to the structural

engineering aspect. But once the financial implications associated with retractable roof

system is confronted, the need for economical structural system is fully realised.

In this chapter, a brief review of a few retractable roof stadiums constructed in different

parts of the world is also presented. Similarly, with regard to structural optimisation,

genetic algorithms are found to be more appropriate in the present study. Hence the

literature pertaining to the two main variants of genetic algorithms, that is, binary coded

genetic algorithms and real coded genetic algorithms is presented. In addition, a brief

summary on retractable roof stadia, unattended portions of work in earlier research works

and the objectives of the present research work are listed.

2.2 Retractable roof structures

(Kassabian, You et al, 1999) had presented a new concept for geometrical design of

retractable roof structures. According to his study, the structures consist of a foldable

lattice beams connected by cylindrical joints, to which covering panels or membranes are

attached. These structures fold towards their perimeter and there are practically no

restrictions to the shape that can be adopted. Solutions to the key problems that have to be

8

solved in the course of the kinematic design of this new type of structure were presented,

including two different ways of connecting them to fixed foundation points while

maintaining their internal degrees of mobility, and how to determine the shapes of the

covering panels to avoid interference during retraction. A preliminary study of the gravity

induced deflections of the type of structure described in this paper, including simulations

with the finite-element package ABAQUS and experiments on a physical model were

carried out. The study showed that this type of structure behaved in a way similar to a

grillage of beams connected by moment less joints. The gravity-induced deflections vary

during the expansion of the structure, as each beam is subject to twisting moments of

increasing magnitude as it rotates towards the centre. Thus, the innermost points of the

structure deflect downwards. Once the structure is fully expanded, various strategies can be

adopted to make it secure under operational loads. For example, some of the joints can be

latched, or the deployment actuators can be driven beyond the point of first ‘collision’, in

order to preload the whole structure and remove the backlash from the joints.

(Thomas Buhl, Jensen et al, 2001) presented the optimal shape of cover plates for circular,

symmetric retractable roof structures based on a grid of multi-angulated rods connected by

revolute joints. It was assumed that these cover plates are co-planar, and hence they are not

allowed to overlap in any configuration; they are also required to form a gap free surface in

the closed configuration. Suitable shapes for the plates are found by formulating an

optimization problem based on suitably defined overlap and gap area functions. The size of

the hinges of the bar structure is considered in the optimization. To stabilize the

optimization convergence various move limit strategies are tested and a min-max

formulation was proposed. The optimization problems were solved by the method of

moving asymptotes and the sensitivities are found by finite differences. A series of test

9

cases were presented, whose results agree with previously known solutions, however the

generality of the present method makes it suitable to develop structures of general shape.

(Noémi Friedman and GyörgyFarkas, 2011) have focused on review of roof structures that

are movable either for enabling quick and/or safe construction or in order to adapt the

structure to external excitations. Roof designs coming from both motives were discussed.

After a short review on historical background an extensive overview was given on

different types of transformable roof structures, namely retractable roofs with rigidly

moving parts, retractable/deployable pantograph structures, the pantadome erection,

deployable tensegrity structures, retractable/deployable membrane structures, pneumatic

structures and constructional methods of concrete shell structures were presented. Early

transformable designs appeared in first place for housing sport venues. With the currently

growing media focus on sport events, the demand for retractable structures is steadily

increasing. Most of these designs use rigid moving parts to retract the roof structure. In

most of the situations the slicing of an ideal roof shape results in gigantic structural

heights. Mechanical instruments enabling retraction increases the costs of the structures

further.

On structural aspects pertaining to retractable roof structure very few research papers are

available as on date. A review of existing retractable roof structures constructed in various

parts of the world is presented here.

Rogers Centre

Rogers Centre is the first full-scale retractable roof stadium of the world. It is located in

Toronto, Ontario, Canada and the construction was completed in 1989 with a cost of about

C$578 million. Then called as Sky dome, but since renamed the Rogers Centre. The

stadium was designed by local architect Rod Robbie, the structural engineer was Mike

Allen and developed by Bill Neish and RAN consortium. The roof rises to a height of 310

10

feet with a span of about 700 feet and has a seating capacity of nearly 50,000 spectators.

The roof is a massive steel structure weighing 22,000,000 pounds, consists of four panels,

the larger one is stationary and the other three are moveable. The last panel rotates 180

degrees to completely close the stadium. The roof is linked to wheeled bogies that roll by

electrically driven train engines with the tracks supported on concrete superstructure and

takes 15 to 20 minutes to open or close.

Figure 2.1 Rogers Centre in closed and open position(http://www.gotherguide.com,

http://hustleanheart.blogspot.com)

Bank One Ballpark

Bank One Ballpark is the first retractable roof stadium in United States located at Phoenix,

Arizona. Desert climate of Arizona necessitated having an air-conditioned retractable roof

stadium. The stadium was designed by Ellerbe Becket and was completed in 1998 at a cost

of $354 million of which $70 million is for retractable roof and has a seating capacity of

about 49,000 spectators. It was developed by the contractors Schuff Steel Company and

was renamed as Chase Field in the year 2005. The roof rises to 200 feet height and spans

517 feet with 6900 ton roof weight. The roof of the stadium consists of two telescoping

sections that bi-part over the middle of the playing field. Each telescoping section

comprises one stationary and three moving panels; when the roof is open the movable

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panels rest above the stationary one. The retractable mechanism constitutes wheels at the

ends of the panels resting on rails and two 200hp motors engaging four miles of steel cable

to retract the roof like a drawbridge so that the roof is opened or closed in about four

minutes.

Figure 2.2 Bank One Ball Park in open and closed position(http://flickriver.com)

Safeco Field

Safeco Field is a baseball stadium located in Seattle, Washington and was opened in 1999.

It was built at a cost of $517 million of which roof costs $67 million and retractable

mechanism costs $14 million. The stadium was designed by architect NBBJ of Seattle and

has a seating capacity of 47,878 spectators. The roof system constitutes three moving

panels with a span of 600 feet and the middle panel rises to a height of 275 feet. The other

two panels roll beneath the middle panel and all the three panels move to the east side of

the stadium. The panels are pulled by cables to open or close the roof that weighs 11,000

ton in 20 minutes. When the roof is covered the stadium is not fully enclosed, it only

covers the stadium like an umbrella so that spectators feel that they are in an open-air

stadium. The open nature of stadium may cause uplift failure due wind pressure and to

avoid that operable lock-down devices are used to tie down the roof towards the supporting

structure. Each panel of the roof is provided with 18-inch dampers to prevent seismic

failure. The huge weight of the roof causes enormous stresses on the wheels that slide on

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the rails. To lower the stresses in the wheels pivot- beam suspension with large number of

wheels is employed. The transporter assembly of the roof system is about 20 feet deep.

Figure 2.3 Safeco Field in full open and partial open

position(http://cookandcheung.ourwedding.com, http://www.snakejazz.com)

Minute Maid Park

Minute Maid Park is a ballpark in Houston, Texas, United States that opened in 2000 and

has a capacity of accommodating about 40,950 spectators. It was designed by the architects

Populous Capacity and roof retraction mechanism was developed by Uni-Systems. The

roof system constitutes three panels of span 580 feet with middle panel of width 250 feet

and end panels each of 125 feet wide slides beneath the middle one.

An independent wheel spring suspension system was used to avoid individual wheels to

take disproportionate portions of the roof load. The transporter assembly of the roof system

is about 6.5 feet deep. Natural lateral forces such as wind and seismic loads cause

unavoidable deflections in the structure. But the avoidable forces such as thermal

expansion and foundation settlement can produce reactions several times greater than those

from natural forces. Since the wheels on which the roof rests are designed for gravity loads

rather than lateral thrust, a lateral release system was included to relieve these horizontal

reactions from the retractable mechanism. With the inclusion of all these systems, Minute

Maid Park cost almost half as much as Safeco Field in total cost($277 million), roof

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cost($32 million) and mechanism cost($7.5 million) for a relatively similar stadium. The

weight of the roof was also brought down to 9,000 tons.

Figure 2.4 Minute Maid Park in full open position(www.WorldStadiums.com)

Miller Park

Miller Park is a ballpark located in Milwaukee, Wisconsin and the construction was

completed in 2001 with a cost of $400 million. The stadium was designed by NBBJ and

Eppsetin Uhen Architects in collaboration with HKS, Inc., and the contractors were from

Mitsubishi Heavy Industries of America. The roof system constitutes fan arrangement of

seven panels with a span of 600 feet and rises to a height of 330 feet, which pivot from a

point to move along a semi-circular track. Three of the panels slide over a fixed panel on

the left and two panels slide over a fixed panel on the right field side. Each panel slides

along a steel rail on a pair of two-wheeled bogies.

The unconventional fan-shaped roof has met with complications; major elements of the

pivot system behind home plate and the outfield roof track have been replaced. At the end

of 2006 season, the roof’s bogie system was replaced at a cost of over $13 million.

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Figure 2.5 Miller Park in paritial open position(http://fineartamerica.com)

Reliant Stadium

Reliant stadium is the first NFL stadium to have retractable roof and is opened in the year

2002. The stadium was designed by architects HOK Sport (Populous since 2009), the

structural engineers were Walter P Moore Engineers and Consultants, and the roof

mechanism was developed by Uni-Systems. The roof system consists of two large panels

with a span of 385 feet and 245 feet width that bi-part the field. Each panel contains five

tapered depth tri-chord trusses each of 30 feet deep rolls along conventional rail assembly

using just 20 wheels and 40-5hp motors and roof can be opened or closed in 10 minutes.

Transporter assembly is less substantial than other retractable roofs since the roof is much

lighter at a little over 1,000 tons. Roof weight is less but supporting structure compensates

with daunting dimensions. Rails on which the panels roll are ultimately carried by two

super trusses, which span 650 feet inside the stadium and an additional 167 feet beyond the

ends to support the open roof. The light roof eliminated the need for a special wheel

suspension and instead used a linked carriage suspension system. The roof itself handles

wind and seismic loads with a lateral release mechanism called 4-bar linkage. The 4-bar

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linkage was designed to connect the ends of the trusses to the transport carriers. As the roof

deflects horizontally, the bars pivot to permit this position but also continue to transfer the

gravity load of the roof down through the wheels with minimal horizontal thrust at the

wheel. The roof can sway up to 21.5 inches in either direction safely. This innovative

solution prevented the flexibility of the structure’s roof from obstructing its operation and

instead created a simple mechanism with little added material.

Figure 2.6 Reliant Stadium in full open and full closed

position(http://blogs.houstonpress.com, www.WorldStadiums.com)

University of Phoenix Stadium

University of Phoenix is a first retractable roof and playing field stadium located in

Glendale, Arizona in America built at a cost of $455 million with seating capacity of

63,400 spectators and is opened in 2006. The stadium was designed by the architect Peter

Eisenman/Populous(HOK Sports) and the structural engineers are from Walter P. Moore.

The shape of the stadium is loosely modeled after a ’barrel cactus’, a wide spread plant in

the Arizona desert and the roof is made of translucent ‘Bird-Air’ fabric. An opening

provided on one side of the stadium allows the playing field to slide to the exterior of the

building, allowing the entire natural turf playing surface to be exposed to sun light when it

is not in use, also allowing the floor to be used for other purposes without damaging the

playing surface.

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The roof has two large retractable panels that cover the entire playing field while providing

maximum shading for spectators. It is the first retractable roof ever built on an incline. The

roof is supported by two 700 feet long trusses which rise to a height of 206 feet above the

ground. Each truss is 87 feet deep at its tallest point and weighs about 1800 tons. The roof

panel each of which weighs 550 tons takes 15 minutes to open. The stadium roof is closed

and air-conditioned during hot months, and is opened in the cooler climate.

Figure 2.7 University of Phoenix Stadium in full open and closed

position(http://commons.wikimedia.org, http://www.stadiumtravelguide.com)

Cowboys Stadium

The Cowboy’s stadium is the largest domed stadium with retractable roof located at

Arlington, Texas built at a cost of $1.3 billion and opened in 2009. The stadium has the

largest column free interior with a seating capacity of 80,000 spectators. The stadium was

designed by architects HKS Inc. and the structural engineers are from Walter P. Moore.

The retractable roof mechanism was developed by Uni-systems and the contractor is

Manhattan Construction Company.

The roof is supported by two 35 feet deep and 15 feet wide boxed arch trusses, each of

span 1290 feet length and 292 feet height weighs 3255 tons is grounded at each end. Two

bi-parting mechanized roof panels each measuring 63,000 sq. ft. will be driven by a rock-

and -pinion drive system consisting of 64-7.5 hp. Electric motors, making it first of its kind

in the world.

17

Figure 2.8 Cowboys Stadium in full open and full closed

position(http://sodacolapepsi.blogspoy.com, http://blogs.dallasobserver.com)

Lucas Oil Stadium

Lucas oil stadium is located at Indianapolis, Indiana and was built in 2008 to accommodate

63000 spectators for football game. HKS Inc. is the architectural firm, Walter P Moore was

the structural engineer of record and the roof mechanism was developed by Uni-Systems.

The stadium has a retractable roof that divides lengthwise into two retractable panels, with

half sliding down the sloping roof of the stadium into the open position. A cable drum

drive system drives the retractable roof panels up and down the sloped track. The roof

boasts the largest opening of 4.5 acre hole to the sky.

Figure 2.9 Lucas Oil Stadium in full open and full closed

position(http://stampedeblue.com, http://www.in.gov)

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Oita Stadium

Oita stadium is in the city of Oita in Kyushu Island in Japan. It was opened in 2001 and has

a seating capacity of 43,000. It was designed by the famous architect Kisho Kurukowa and

built by KT Group, Takenaka Corporation. The stadium has a retractable dome roof with

roof system driven by a wire traction system. The total cost of construction of the stadium

was ¥25 billion.

Figure 2.10 Oita Stadium in full open and full closed position(http://www.info-stades.fr,

http://www.footballzz.com)

Toyota Stadium

Toyota stadium is a retractable roof stadium in Toyota, Japan, built in 2001 with a seating

capacity of 45,000. It was designed by architect Kisho Kurokawa. The roof contains an

accordion-like moving element that is operated by a system of air-pillows that enable the

stadium to completely close itself.

Figure 2.11 Toyota Stadium in full open position and inner view of

Stadium(http://www.WorldStadiums.com, http://www.skyscrapercity.com)

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Wembley Stadium

Wembley stadium located at Borough of Brent, London, opened in 2007, is the second

largest stadium in Europe with a seating capacity of 90,000. Designed by Foster and

Partneees, HOK sport, and the structural engineer was Mott MacDonald. A signature

feature of the stadium is the 134 m. high Wembley arch with a span of 317 m. This the

longest single span steel arch retractable roof structure in the world. The stadium was built

by Australian firm Multiplex at a cost of £798 million.

Figure 2.12 Wembley Stadium in full open position with its inner

view(http://www.progressivegroup.co.uk, http://www.totalprosports.com)

2.3 Structural Optimization

The structural design optimisation of retractable roof structure has not been attempted by

any researcher till date. The reason could be due to the fact that the structural design of

retractable roof structure is no different from other types of structures. However, since the

high cost of the structure is the discouraging aspect of retractable roof structures, the study

of structural optimisation is primarily carried-out here.

As explained in section 1.6, the optimization methods are broadly divided into two types as

classical methods and methods based on evolutionary algorithms. The classical method

called mathematical programming is used for structural optimization. Structural

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optimization using mathematical programming is once again is categorized as linear

programming and nonlinear programming. In linear programming the objective functions

and the constraints are represented as linear combination of design variables. The analysis

can’t be error free when a linear relationship is used to model a nonlinear structural

response. Nonlinear mathematical programming uses the Kuhn-Tucker condition for

optimizations, which is extremely difficult for most structural optimization problems. In

addition to mathematical programming, optimality criteria methods were presented by

many earlier researchers, which are useful for continuous design variables problems and

are difficult to use for large scale discrete variable problems.

2.4 Genetic Algorithms

2.4.1 Binary Coded Genetic Algorithm

(Goldberg, 1989) has applied a simple genetic algorithm to the optimization of a 10-

member plane truss problem. The objective of the problem was to minimize the weight of

the structure subjected to stress constraints on each member. A three operator genetic

algorithm consisting of Roulette Wheel selection, simple crossover and mutation was used

with constraints adjoined using a quadratic external penalty function. The design variables

are the ten member areas, were coded as a concatenated, mapped fixed- point string where

each of the 10 four bit area mapped linearly between minimum and maximum value of the

area of cross-section in the domain.

(Rajeev and Krishnamoorthy, 1992) have presented a simple binary coded genetic

algorithm for optimizing structural systems with discrete design variables. The genetic

algorithm presented here was a modified simple genetic algorithm proposed by Goldberg.

As genetic algorithm are best suited for unconstrained optimization problems a penalty

based transformation method is used to convert it into a constrained problem. In their

21

study, two genetic operators i.e. reproduction and crossover operators were used in

implementation of genetic algorithm. A few standard truss optimization problems were

solved and the results are compared. To illustrate the applicability of the genetic algorithm,

a relatively large problem of a 160-transmission tower was solved and the results were

presented.

(Adeli and Cheng, 1994) presented an augmented Lagrangian binary coded genetic

algorithm for optimization of structures utilizing the multiprocessing capabilities of the

high performance computers. Two different concurrent genetic algorithms for optimization

of large structures utilizing the inherent parallelism of genetic algorithms were presented.

One strategy was to parallelize the algorithm among the strings in the entire population,

that is, to assign all the fitness-function evaluation, and crossover and mutation

computations to the available number of processors equally. Since all of these

computations for each string are independent of other strings, the algorithm can be fully

parallelized within these three stages. In this stage, the most time consuming step was

divided into a number of data-independent portions equal to the number of available

processors. The second strategy was to initially divide the population equally into a

number of subpopulations equal to the number of available processors. Employing the

augmented Lagrangian genetic algorithm, each processor performs optimization in its own

subpopulation independently. Each subpopulation has its own penalty-function coefficients

and these coefficients are updated by each processor independently of other processors.

After a limited number of iterations or the stopping criterion is met for all the processors,

the subpopulations were combined. Then another stopping criterion was checked for the

entire population. If this criterion is met, the optimum solution is found and the processor

is terminated. It was observed that the performance of both algorithms improves with the

size of the structure.

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(Jiaping Yang and Chee Kiong, 1997) presented an approach to optimization design

concerning the configurations of structures using genetic algorithm with a tournament

selection strategy. The tournament selection strategy was used as a replacement for the

commonly used fitness-proportional selection strategy to drive the genetic algorithm so as

to improve the fitness of each succeeding generation more efficiently. Numerical results

for three examples reveal that a significant reduction of computation cost was achieved in

the proposed genetic algorithm with tournament selection, as compared to the widely used

genetic algorithm with fitness-proportional selection.

(Rajeev and Krishnamoorthy, 1997) presented a variable string length binary coded genetic

algorithm based methodologies for obtaining optimal design solutions simultaneously

considering topology, configuration and cross-sectional parameters in a unified manner.

The two-phase method presented can handle both discrete as well as continuous design

variables. Configuration optimization problems were solved using this method, in which

size variables were considered as discrete and configuration variables (position of joints) as

continuous. The variable string length used was taken into account variations in topology

in addition to size and configuration. Need for consideration of all the three types of

variables in design optimization of trusses is illustrated by solving a large practical

problem.

(Charles Camp, Pezeshk et al, 1998) presented a design procedure incorporating a simple

genetic algorithm for discrete optimization of two-dimensional structures. The genetic

algorithm based design procedure FEAPGEN was developed as a module in the Finite

Element Analysis Program (FEAP). The special features of FEAPGEN includes discrete

design variables, an open format for prescribing constraints, design checking using the

American Institute of Steel Construction Allowable Stress Design (AISC-ASD)

specifications, multiple loading conditions, and a comprehensive AISC database of

23

available structural steel members. Several strategies for reproduction, crossover and

penalty functions were investigated for their appropriateness to the ASD design of two-

dimensional structures.

(Erbatur, Hasancebi, et al, 2000) reported development of a computer–based systematic

approach for discrete optimal design of planar and space structures composed of one-

dimensional elements using a genetic algorithm as the optimizer. It was observed that a

genetic algorithm often finds the region of the search space containing the global optimum,

but not the true optimum itself. In this study an approach based on multilevel optimization

is tested and proved to overcome this shortcoming. The approach rests on reducing the size

of the search space for individual design variables in each successive level of the

optimization process.

(Kamal C. Sarma and Adeli, 2000) presented a fuzzy augmented Lagrangian genetic

algorithm for optimization of steel structures subjected to the constraints of the AISC

allowable stress design specifications taking into account the fuzziness in the constraints.

The membership function for the fuzzy domain was found by the intersection of the fuzzy

membership function for the objective function and the constraints using the max-min

procedure of Bellman and Zadeh. Nonlinear quadratic fuzzy membership functions were

used for objective function and constraints. The features and advantages of the new fuzzy

genetic algorithm include acknowledging the imprecision and fuzziness in the code-based

design constraints, increased likelihood of obtaining the global optimum solution,

improved convergence and reduced total computer processing time.

(Pezeshk, Camp et al, 2000) presented a genetic algorithm based optimization procedure

for the design of 2D, geometrical, nonlinear steel-framed structures. The approach

presented uses genetic algorithm as a tool to achieve discrete nonlinear optimal or near-

24

optimal design. Frames were designed in accordance with the requirements of the AISC-

LRFD specification. This paper employs a group selection mechanism, discusses an

improved adapting crossover operator and provides recommendations on the penalty

function selection. The differences between optimized designs obtained by linear and

geometrically nonlinear analysis were compared.

(Kamal C. Sarma and Adeli, 2001) reported optimization of very large steel structures

subjected to the actual constraints of the American Institute of Steel Construction ASD and

LRFD specifications on high-performance multiprocessor machines using biologically

inspired genetic algorithms. First, parallel fuzzy genetic algorithms were presented for

optimization of steel structures using a distributed memory Message Passing Interface

(MPI) with two different schemes: the processor farming scheme and the migration

scheme. Next, two bi-level parallel genetic algorithms were presented for large-scale

structural optimization through judicious combination of shared memory data parallel

processing using the OpenMP Application Programming Interface (API) and distributed

memory message passing parallel processing using MPI.

A flexible binary coded Genetic Algorithm Library using object-oriented approach was

developed to address the problems of practical optimization of large space trusses by

(Krishnamoorthy, PrasannaVenkatesh et al, 2002). It is shown how classes derived from

the implemented libraries can be used for optimization problems, where several

constructability aspects have been incorporated to simulate real-world design constraints.

Strategies were also suggested for member grouping for reducing the problem size and

implementing move-limit concepts for reducing the search space adaptively in a phased

manner. The implemented libraries are tested on a number of previously fabricated space

trusses and results were compared.

25

(Lee and Ahn, 2003) used a genetic algorithm to perform discrete optimization of

reinforced concrete plane frames subjected to combinations of gravity loads and lateral

loads. Difficulties in finding optimum sections from a semi-infinite set of member sizes

and reinforcement arrangements were alleviated by constructing data sets, which contain a

finite number of sectional properties of beams and columns in a practical range.

Construction practice was also implemented by linking columns and beams by group and

by considering connectivity between columns located in the same column line. Different

convergence trends were observed when different values for the parameters were used. For

cases with fixed mutation rate, convergence rate decreased and showed more variations as

the mutation rate increased. For the problems considered, increasing the gravity load and

decreasing the mutation rate gradually as generations proceeded resulted in a better design

than in other cases. It was shown that the developed genetic algorithm obtained an optimal

design for reinforced concrete plane frames.

(Sivakumar, Rajaraman et al, 2004) presented an object-oriented optimization approach for

steel lattice towers using genetic algorithm. The lattice tower was considered as a

collection of separate objects of panels as they possess an independent property as well as

inherent properties. This can considerably reduce the design space of the problem and

enhance the result. The objects were optimized separately and combined to obtain an

improved and efficient solution.

(VedatTogan and Daloglu, 2005) presented adaptive approach in genetic algorithm to show

how the adaptive approach affects the performance of genetic algorithm, suggesting some

improvements in both the penalty function, and mutation and crossover. A strategy was

also considered for member grouping to reduce the size of the problem. It was concluded

that the member grouping together with the adaptive approach increase the probability of

catching the global solution and enhance the performance of genetic algorithm.

26

(Pandia Raj and Kalyanaraman, 2005) have demonstrated application of genetic algorithm

for optimum design of truss type railway bridges. The main thrust of this paper was to

illustrate typical application of genetic algorithm to practical design of structural systems

such as steel truss girder railway bridge superstructures in an object oriented

implementation of design and optimization codes. In this work binary string for coding of

truss girder design variables, fitness as a ranking measure of the adaptability to the

environment, selection criteria and genetic operators such as crossover and mutation were

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.

(VadatTogan and Daloglu, 2009) presented a minimum weight design of steel bridge

trusses under the limitations imposed on its behaviour by the design codes when it was

subjected to moving loads. Adaptive manner operators were used to enhance the capability

of genetic algorithm. Design examples were solved to illustrate the applicability of

proposed algorithm and the results were compared with those evaluated by various

optimization methods with continuous design variables such as sequential convex

programming (SCP), sequential quadratic programming (SQP) and evolution strategy

(EVOL). It was concluded that the results obtained by genetic algorithm was meaningful,

more suitable for practice and genetic algorithm performs well to find minimum weight of

the bridge trusses under moving load for optimization with discrete design variables.

(Al-shihri, 2010) has presented a novel evolutionary algorithm based upon genetic

algorithm which is suitable for a general class of structural optimization problems. The

algorithm was applied for discrete and/or continuous type(s) of design variables. Proposed

algorithm was designed such that it converges rapidly to local optima whenever a local

optimum solution is nearby. In each generation, the algorithm selects a chromosome from

27

the population that represents a design close to a local optimum. Further, a new set of

chromosomes called Single Digit Chromosome (SDC) were generated having all zero bits

except for one. Chromosomes are selected from the population and their binary addition

and subtraction are performed with the SDCs. Through a number of test examples it was

shown that the proposed algorithm is much robust and reliable as compared to the

traditional genetic algorithm.

2.4.2 Real coded Genetic Algorithm

(Deb and Beyer, 1999) have demonstrated the self-adaptive feature of real parameter

genetic algorithm using simulated binary crossover (SBX) operator and a mutation

operator. The connection between the working of self-adaptive Evolutionary Strategies and

real-parameter genetic algorithm with SBX operator was also discussed. Thereafter, the

self-adaptive behaviour of real-parameter genetic algorithms is demonstrated on a number

of test problems commonly-used in the Evolutionary Strategies literature. The remarkable

similarity in the working principle of real-parameter genetic algorithms and self-adaptive

Evolutionary Strategies shown in this study suggests the need of emphasizing further

studies on self-adaptive genetic algorithms.

(Deb, Ashish et al, 2002) have explained that due to increasing interest in solving real-

world optimization problems using evolutionary algorithms (EAs), researchers have

recently developed a number of real parameter genetic algorithms (GAs). In these studies,

the main research effort was spent on developing an efficient recombination operator. Such

recombination operators use probability distributions around the parent solutions to create

an offspring. Some operators emphasize solutions at the centres of mass of parents and

some around the parents. In this paper, they proposed a generic parent-centric

recombination operator (PCX) and a steady-state, elite-preserving, scalable, and

28

computationally fast population alteration model (called as G3 model). The performance of

the G3 model with the PCX operator was investigated on three commonly used test

problems and was compared with a number of evolutionary and classical optimization

algorithms including other real-parameter GAs with the unimodal normal distribution

crossover (UNDX) and the simplex crossover (SPX) operators, the correlated self-adaptive

evolution strategy, the covariance matrix adaptation evolution strategy (CMA-ES), the

differential evolution technique, and the quasi-Newton method. The proposed approach

was found to consistently and reliably perform better than all other methods used in the

study. A scale-up study with problem sizes up to 500 variables shows a polynomial

computational complexity of the proposed approach. This study demonstrates the power of

the proposed technique in tackling real-parameter optimization problems.

(Camp, Pezeshk et al, 2003) reported a design procedure implementing a genetic algorithm

developed for discrete optimization of reinforced concrete frames (RC-GA). The design

procedure confirms to the American Concrete Institute (ACI) Building Code and

Commentary. The objective of the RC-GA procedure was to minimize the material and

construction cost of reinforced concrete structural elements subjected to serviceability and

strength requirements described by the ACI Code. Beam elements were evaluated based on

their flexural response considering moment magnification factors due to frame stability. A

rectilinear column strength interaction diagram was used to evaluate the feasibility of

columns with moment magnification due to slenderness effects. The limitations and

specifications of the ACI Code were formulated as a series of constraints to the discrete

cost optimization problem and applied as penalties on the fitness function of the genetic

algorithm. Examples were presented demonstrating the efficiency of the RC-GA

procedure for the flexural design of simply supported beams, uniaxial columns and multi-

story frames.

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(Herrera, Lozano et al, 2005) presented a model of hybrid crossover operator as a suitable

tool to facilitate the study of the synergy amongst real-parameter crossover operators with

different search biases. They generate two offspring using two crossovers chosen from

different groups of taxonomy. Hybrid crossover operators achieve a finer performance than

homogeneous crossover operators. Thus, the hybridisation of real-parameter crossover

operators shows promise as a strategy to improve the effectiveness of this genetic operator.

The joint application of BLX-α (α=0.5) and dynamic heuristic crossover induces an

appropriate relationship between exploration and exploitation to produce profitable

synergic effects, allowing a robust operation to be achieved for test functions with different

characteristics.

(Xinping Liu and Ying Liu, 2009) described that population diversity was the precondition

of population evolution. Premature convergence is a problem in genetic algorithm. It is

seriously influenced by the distribution properties of initial population. So an adaptive

genetic algorithm based on diversity was proposed in their paper. It uses the max-min

distance means and makes various individuals maintain certain Hamming distance to

produce good population distribution. Simultaneously the genetic operators were

adaptively determined according to the population diversity and individual fitness. The

diversity of population was effectively maintained and a global optimal solution was

quickly obtained using the proposed method. Finally, four representative test functions

were chosen to test the improved adaptive genetic algorithm's capability. The simulation

and comparison results show the validity of this algorithm.

(Carlos ConceiçãoAntónio, 2009) have shown that efficient combination of multiple

crossover operators in structural optimization can produce important synergy effects

improving the performance of Genetic Algorithms. In particular this concept was explored

through the Hierarchical Genetic Algorithm (HGA) that results from application of the

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combination of hybrid crossover operators and heterogeneous distributed GAs. In order to

study the synergy effects six crossover strategies were considered. These strategies were

built using several combinations of the following crossover operators: elitist hybrid

crossover with genetic improvement (EHCgi), elitist parameterised uniform crossover

(EpUC) and age parameterised uniform crossover (ApUC). The use of multiple crossovers

was implemented and tested at both sub-population level and hierarchical topology level.

Considering two examples of structural optimisation different styles of exploitation and

exploration of the design space of the crossover strategies show different synergy effects as

follows:

Synergy effects were higher when pure elitist crossover strategies were considered;

Crossover strategies based on mixed elitism and age-structured populations show

synergy towards the end of the evolutionary processes;

Synergies depend on the ability to build an appropriate strategy based on the use of

multiple crossovers;

High values of crossover replacement rate in elite group were important to obtain

considerable synergy effects;

It was possible to obtain good synergy effects without using purely elitist strategies.

From the numerical examples it was concluded that synergy was revealed when multiple

crossover operators were used. This was important for the user/designer that can leverage

the existing synergy searching substantial improvements in GA usage. So, in order to

interpret the meaning of synergy and when and how it might be obtained some guidelines

were established as follows:

The modularity of HGA facilitates the combination of hybrid crossover operators

and heterogeneous distributed GAs with benefits for the relationship between

exploration and exploitation inducing important synergetic effects.

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Hybridization is recommended as a promising strategy to improve the effectiveness

of crossover operators.

It is possible to reach synergy by using crossover operators from different

taxonomy groups. The herein described combination of neighbourhood based

crossover operator (NBCO) together with discrete crossover operators (DCO)

enhances this feature.

The joint application of apparently contradictory evolutionary concepts such as

pure elitism and species conservation paradigm is possible. The combination of

elitist strategies together with age parameterised uniform crossover (ApUC) shows

synergetic improvements. However, some ability is required so that

complementarities between exploration and exploitation of space design search can

be achieved.

They concluded that, the use of multiple crossover operators and their hybridization was

promising and necessary to obtain important synergies.

(Rituparna Datta, Deb et al, 2010) have used an adaptive mutation based Real-Coded GA

(RGA), which uses a popular penalty parameter-less approach to handle constraints and

search the feasible region effectively for the global best solution, and at the same time use

an adaptive mutation strategy to maintain diversity in the population to enable creation of

new solutions. They have coupled RGA with ideas from the gradient projection method to

specifically handle equality constraints. They have found our simple procedure working

quite well in most of the test problems provided as part of the competition on Single-

objective Constrained Real Parameter Optimization.

(Oliver Kramer, 2010) stated that the success of evolutionary search depends on adequate

parameter settings. Ill conditioned strategy parameters decrease the success probabilities of

genetic operators. Proper settings may change during the optimization process. The

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question is whether adequate settings can be found automatically during the optimization

process. An evolution strategy for online parameter control problem is self-adaptation.

Self-adaptation is the implicit search in the space of strategy parameters. The self-adaptive

control of mutation strengths in evolution strategies turned out to be exceptionally

successful. Nevertheless, for years self-adaptation has not achieved the attention it

deserves. The author has carried out a survey of self-adaptive parameter control in

evolutionary computation. It classifies self-adaptation in the taxonomy of parameter setting

techniques, gives an overview of automatic online-controllable evolutionary operators and

provides a coherent view on search techniques in the space of strategy parameters. Beyer

and Sendhoff’s covariance matrix self-adaptation evolution strategy was reviewed as a

successful example for self-adaptation and exemplarily tested for various concepts. For

each type of strategy parameter adequate operators for the exploration of the strategy

variable search space have been proposed. The best results for self-adaptive parameters

have been achieved for mutation operators. Most theoretical work on self-adaptation

concentrates on mutation. A necessary condition for the success of self-adaptation is a tight

link between strategy parameters and fitness: if the quality of the search process strongly

depends on a particular strategy variable, self-adaptive parameter control is reasonable. But

not every parameter type offers a strong link. e.g., crossover points suffer from a weak

link.

(TayfunDede, Bekiroglu et al, 2011) studied the effect of encoding types such as value

encoding and binary encoding for continuous and discrete optimization with a genetic

algorithm, which was coded in FORTRAN and considers stress and displacement

constraints, in view of weight minimization of truss structures. Moreover, when continuous

optimization was performed, the challenge of huge search space due to the effort of

considering continuous set of design variables was overcome by a mechanism introduced

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as restricted range approach (RRA) in the study. In value encoding, when the crossover

operation was carried out, the fit chromosome was never lost, but in the binary encoding,

the crossover can destroy the fit chromosome, and a new chromosome with low fitness can

occur after this operation. It was also observed that the binary encoding may require a large

number of genes to change a chromosome when a small change in the parameter was

needed. In comparison with the literature, it was concluded that the program developed in

this study can be effectively used in the weight minimization of truss structures. It was also

come to the conclusion that value encoding overcomes the adverse effects of Hamming-

cliff, and that value encoding requires less computer memory and time, never destroys the

fit chromosomes.

(Su, Yan et al, 2011) have developed a generalized genetic algorithm based on the

inbreeding criterion and heterogeneous strategy to find the global optimal reinforcement

contents for a concrete solid structure subjected to a general three-dimensional (3D) stress

field. The inbreeding criterion was used to judge the occurrence of inbreeding, whereas the

heterogeneous strategy was used to eliminate the inbreeding individuals. Many effective

techniques such as the population isolation mechanism, optimum reserved strategy and

arithmetic crossover were also adopted in the computations. It was demonstrated from the

solution of the optimization problem that the improved generalized genetic algorithm

(GGA) was suitable for solving complicated optimization problems. The reinforcement

design method based on the improved GGA can find the global optimal design solution of

a general reinforced concrete element. According to the authors, the method provides the

designer with a valuable tool for the dimensioning of reinforcements in concrete solid

structures.

(Otman and Jaafar, 2011) have discussed the basic conceptual features and specific

characteristics of various crossover operators in the context of the Traveling Salesman

34

Problem (TSP) so as to develop a robust genetic algorithm. The authors implemented six

different crossover procedures and their modifications in order to test the influence of the

recombination operators to the genetic search process when applied to the traveling

salesman problem. The following crossover operators have been used in the

experimentation: the Uniform Crossover Operator (UXO), the Cycle Crossover (CX), the

Partially-Mapped Crossover (PMX), the Uniform Partially-Mapped Crossover (UPMX),

the Non-Wrapping Ordered Crossover (NWOX) and the Ordered Crossover (OX). The

results of experimental comparison of more than six different crossover operators for the

TSP were presented. The experiment results show that OX operator enables to achieve a

better solution than other operators tested. According to the comparative study of the

crossover operators mentioned by them, the development of innovative crossover operators

for the traveling salesman problem may be the subject of the future research.

(Himanshu and Deb, 2011) expressed that most real-parameter genetic algorithms (RGAs)

use a blending of participating parent solutions to create offspring solutions through its

recombination operator. The blending operation creates solutions either around one of the

parent solutions (having a parent-centric approach) or around the centroid of the parent

solutions (having a mean-centric approach). In their paper, they argued that a self-adaptive

approach in which a parent-centric or a mean-centric approach was adopted based on

population statistics is a better procedure than either approach alone. The authors proposed

a self-adaptive simulated binary crossover (SA-SBX) approach for this purpose. On a suite

of eight uni-modal and multi-modal test problems, they demonstrate that a RGA with

SASBX approach performs reliably and consistently better in locating the global optimum

solution than the RGA with the original parent-centric SBX operator and the well-known

CMA-ES approach.

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(Fakhir, Mustafa et al, 2012) have developed a FORTRAN program which uses the finite

element methods based numerical analysis and was modified to achieve size optimization

based on genetic algorithm to perform the analysis and design of the space trusses. The

algorithm searches all the available solution under the constraints of allowable stress and

displacement so as to find the best solution. The best solution was selected among all

available results satisfying the constraints by the algorithm. The design variables were

cross sectional dimensions, while the solution was done for various types of sections. For

all design variables significant decrease in weight of material with respect to the stress and

displacement constraints were obtained.

(Azad, Kulkarni et al, 2012) described a mutation-based real-coded genetic algorithm,

MBRCGA, for sizing and layout optimization of planar and spatial truss structures under

stress, displacement and buckling constraints. The Gaussian mutation operator was used to

create two mutation operators of the proposed MBRCGA. The standard deviation needed

for each operator was adaptively adjusted by the population itself. A global search in the

initial iterations was considered, which gradually leads to a local tuning in the last

iterations of the optimization process. In order to handle constraints, an adaptive penalty

function was proposed to reduce the disadvantages of using static penalty constants. In the

selection stage the tournament selection operator was used with an adaptive tournament

size in order to adjust the balance between exploration and exploitation. The performance

of the proposed method was investigated in five typical weight minimization problems of

planar and spatial truss structures with both discrete and continuous design variables.

Optimum designs found by MBRCGA were compared to the recently reported results in

the literature. The results indicate the efficiency, reliability and robustness of the proposed

MBRCGA.

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(Debayan Deb and Deb, 2012) described that the mutation is an important operator in

genetic algorithms (GAs), as it ensures maintenance of diversity in evolving populations of

GAs. Real-parameter GAs (RGAs) handles real-valued variables directly without going to

in a binary string representation of variables. Although RGAs were first suggested in early

nineties, the mutation operator is still implemented variable wise and independently for

each variable. In this paper, they investigate the effect of five different mutation schemes

for RGAs for two different mutation operators. Based on extensive simulation studies, it

was observed that a mutation clock implementation was computationally quick and also

efficient in finding a solution close to the optimum on four different problems used in this

study for both mutation operators. Moreover, parametric studies with their associated

parameters reveal suitable working ranges of the parameters. Interestingly, both mutation

operators with their respective optimal parameter settings were found to possess a similar

inherent probability of offspring creation, a matter that is believed to the reason for their

superior working. This study signifies that the long suggested mutation clock operator

should be considered as a valuable mutation operator for RGAs.

(Aminifar, Aminifar et al, 2013) have investigated the application of an augmented genetic

algorithm (AGA) for the problem of optimal design of truss structures. The problem

studied was to determine the minimum value of the weight/cost associated with the truss

structure design while a set of stress and displacement constraints were to be satisfied. The

proposed AGA exploits a probabilistic selection procedure based on the annealing process.

Moreover, a new enhancing trick was proposed that prevents familial reproduction. This

effect restrains the degenerative phenomena during the evolution of the canonical genetic

algorithm (GA). Accordingly, a considerable improvement in converging speed was

achieved. This advancement could be extremely advantageous in large structure design

problems, where the existing methods suffer from long execution times. Various

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benchmark examples were examined to demonstrate the performance of the proposed

method. Moreover, the obtained results were compared with those of existing methods to

verify the effectiveness of AGA optimization.

(Rajesh Kumar, 2013) presented the simultaneous cost, topology and standard cross-

section optimization of single storey industrial steel building structures. The considered

structures were consisted from main portal frames, which were mutually connected with

purlins. The optimization was performed by the Genetic algorithm (GA) approach. The

proposed algorithm minimizes the structure’s material and labour costs, determines the

optimal topology with the optimal number of portal frames and purlins as well as the

optimal standard cross-sections of steel members.

2.5 Summary of information collected on retractable roof stadia

1. In the beginning the government agencies were reluctant to give permission to the

construction of retractable roof structures as their technology was new hence

considered risky. The financial companies too were unwilling to finance

construction of such unproven high risk structures.

2. In spite of the huge financial burden, the Rogers Centre is still acknowledged as a

magnificent structural accomplishment.

3. With the remarkable success of Reliant Stadium, the popularity of retractable roof

stadia has received wider acceptance.

4. On the automation front, the retractable roof structure technology has advanced

considerably in recent years. Very large size TV screens accompanied by powerful

audio systems are also part of these stadiums to give a boost to the sports activities.

5. However, a formalised approach to be adopted for making the retractable roof

structures is not tabled anywhere to refer for guidance.

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6. Similarly, as on date neither some stipulated safety measures nor standard codes of

practice for 1) material specifications, 2) acoustic & lighting specifications, 3)

disaster prevention mechanisms to be adopted and 4) methods of construction are

available.

2.6 Issues not addressed in the earlier works

1. The focus of the earlier research on retractable structures was restricted mostly to

the development of appropriate geometry only. No attempt was made earlier to

optimize these structures.

2. Earlier researchers had used commercially available softwares for the design of

retractable roof structures. And, none of this commercially available structural

analysis software supports structural optimization not to mention of genetic

algorithms.

2.7 Summary of information collected on Genetic Algorithms

Review of previous research carried-out on application of genetic algorithms to

structural optimization revealed following information:

1. Earlier researchers had implemented genetic algorithms using different types of

genetic operators such as selection operator, crossover operator and mutation

operators. The following were the strategies adopted in the earlier works:

a) Development of new variants of selection operator, crossover operator

and/or mutation operator.

b) Use of hybrid crossover operator by combining multiple crossover

operators.

c) Use of different types of penalty functions.

2. Two variants of genetic algorithms as mentioned below were followed in the past

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a) Binary coded genetic algorithm

b) Real coded genetic algorithm

3. Genetic algorithm with hybrid mutation operator was never attempted in the past

for structural optimization.

Now the onus is on developing algorithm to perform structural analysis and design

optimization using genetic algorithms customized for retractable roof structures.

2.8 Objectives of the present work

Keeping the above points in view, following objectives are set forth for the present thesis:

a) Development of an algorithm that integrates structural analysis and optimal

structural design using real coded genetic algorithm with hybrid mutation

operator.

b) Validation of the developed algorithm with several benchmark problems.

c) Development of algorithm to interactively generate the geometry of retractable

roof structure.

d) Optimal design of retractable roof structure using the validated algorithm.