Adedeji, A A USEP, Journal of Research Information in... and Ejeh, S. P.

Adedeji, A A USEP, Journal of Research Information in Civil Engineering, Vol. 3, No.1, 2006 and Ejeh, S. P.

S I I M P L I I F Y I I N G D E S I G N O F T R U S S S Y S T E M U S I I N G

G E N E T I I C A L G O R I T H M S

*A A Adedeji and S. P. Ejeh

*Department of Civil Engineering, University of Ilorin. PMB 1515, Ilorin,

Nigeria. Email: amadeji@yahoo.com

Department of Civil Engineering, Ahmadu Bello University, Zaria, Nigeria

Abstract

Genetic Algorithm is a powerful search tool enabling an optimum objective function to converge rapidly. Here, two variable formations were used, but functions of more variables were easily incorporated and implemented. The use of binary coding, in a Genetic Algorithm procedure indicated a simple application of this method to a wide variety of optimization problems in design. However, the optimum value of cost function of the truss members was obtained by minimizing the section of the members, which was later compared favourably to and more economically viable than the values obtained from BS 5950 design results.

Keywords

Design, truss system, genetic algorithm, optimization, cost function

1. Introduction

The design of a system is a major field of the engineering profession. The process of designing a system result in a set of drawing, calculations and reports, and the system can be fabricated based on these. Design is an iterative process. The designer’s experience, intuition and ingenuity are required in the design of a system. The use of iteration implies analyzing several trial systems in a sequence before an acceptable design is obtained.

The analysis problem is concerned with determining behaviour of an existing system, or a trial system being designed for a given task. Determination of behaviour of an engineering system is by calculating its response under the specified input (loads). In the highly competitive world of modern technology, it is no longer sufficient to design a system that performs the required task satisfactorily, but it is essential to design an efficient, versatile, unique and cost effective system. To design such system, proper analytical, experimental, and numerical tools are needed.

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It is therefore a challenge for an engineer to design efficient and cost effective systems without compromising their integrity. The conventional design process depends on the designer’s intuition, experience, and skill.

This overwhelming presence of human element can sometimes leads to erroneous and uneconomical results in the synthesis of a complex system.

Furthermore, the conventional design process can involve a lot of calendar time. The optimal design process forces the designer to identify explicitly a set of design variables, a cost function to be minimized, and the constraint functions for the system. Proper mathematical formulation of the design problem is a key to good solutions. Optimization process is more organized by using rigorous information to make decision, and it has been effective and leads to saving construction material during the design process.

The aim of this project is to design a roof truss using Genetic Algorithms to optimize the design variables. In order t o obtain the optimum value of cost function of the material, section of the members was minimized. The results were compared with the conventional method of design. The Pratt type of roof truss was adopted using steel roof truss system for a lecture theater room. Axial forces of the truss system model were computed by using the method of joint resolution. Design consideration to BS 5950 was adopted, while Genetic Algorithm operators such as the selection, cross over, and mutation to the selected structure was employed for the search of optimal design

2. Evolution Algorithms and Search Types

Optimal design process has various conventional methods of computation such as Graphical Optimization, Simpson Optimization Method (Arora,

1989) and of recent the Evolutionary Algorithms. Evolutionary algorithms is an umbrella term used to describe computer-based problem solving systems which use computational model of evolutionary processes as key elements in their design and implementation. A variety of evolutionary algorithms have been proposed. The major ones are: Genetic Algorithms, Evolutionary

Programming, Evolution Strategies, Classifier System and Genetic

Programming. They all share a common conceptual base of simulating the evolution of individual structure via processes of selection, mutation and reproduction. Genetic algorithms method of optimization is adopted in this project

Genetic Algorithms (GAs) are search procedure that uses the mechanics of natural selection and natural genetics (Goldberg 1989). It uses evolutionary

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Adedeji, A A USEP, Journal of Research Information in Civil Engineering, Vol. 3, No.1, 2006 and Ejeh, S. P. techniques, based on function optimization and sometimes artificial intelligence, to develop a solution. Genetic Algorithm (GA) differs from conventional search techniques in several ways. For example, it works with a coding of solution set not the solutions themselves. Also searches are done from a population of solutions and not from a single solution. GA users do not rely on information (fitness function) such as derivatives or other auxiliary knowledge. It uses probabilistic transition rules and not deterministic rules as in conventional methods.

Engineering systems are designed to perform specific tasks. Usually, many alternative designs can perform the same tasks. The important thing is to look for the alternatives that are the best. Economic problems are integral part of engineering because engineers are sensitive to the direct cost of a design.

They must anticipate maintenance and operating cost.

3. History of Genetic Algorithms

The observation of Genetic Algorithms was first mathematically formulated by Holland (1975). Usually the algorithm breeds a predetermined number of generations; each generation is populated with a predetermined number of fixed length binary strings. Holland’s book published in 1975 is generally acknowledged as the beginning of the research of genetic algorithms. Until the early 1980s, the research in genetic algorithms was mainly theoretical with few real applications. De Jong’s (1975) work attempted to capture the features of the adaptive mechanisms in the family of genetic algorithms that constitute a robust search procedure.

Holland (1992) work on adaptive systems laid the foundation for later development; most notably, he was also the first scientist to explicitly propose crossover and other recombination operators. These foundational works established more widespread interest in evolutionary computation. By the early to mid 1980s, genetic algorithms were being applied to a broad range of subjects.

4. Research Methodology

The General Principles of Truss Design to BS 5950: Part I was used in the analysis of the truss system. It was assumed that the members are joined together by smooth pin and the loads consist of concentrated forces applied at the joints. The loads were assumed to be applied at the node points of the members, so that they are principally subjected to direct stresses was used for structural analysis by the joint resolution method.

35


Based on limit state philosophy for building trusses according to BS 5950, the members were designed at the ultimate limit state for strength, stability and fatigue where applicable, and at the serviceability limit state for deflection and durability. Tension members need only be designed at the ultimate limit state.

The compressive strength P

C

depends on the slenderness and the design strength P y

. Tests on axially loaded, pin ended struts show that their behaviour can be represented by a number of curves which relate to the type of section and the axis of buckling. These curves are dependent on material strength and the initial imperfections, which affect the inelastic behaviour and the inelastic buckling load. The appropriate table (Table 25 of BS 5950) was chosen by reference to the section type and thickness, and the axis of buckling. The compressive resistance P

C is,

P c

= A g p cs

(1)

P c

= A g p c for slender sections.

(2) for all other sections. where A g

is the gross sectional area, p cs

is the compressive strength based on a reduced design strength.

The tension capacity of a member P t

= A e p y

in which A e

is the effective sectional area. Where a member is connected eccentrically to its axis then allowance should be made for the resulting moment. Alternatively, such eccentric effects may be neglected by using a lower value of the effective A e

.

For single angle connected through one leg,

3a

1

A e

= a

1

+ a

2

(3a

1

+ a

2

)

(3) where a

1 is the net sectional area of the connected leg and a

2

is the sectional area of the unconnected leg.

5. Problem Formulation

The design of most engineering systems is a fairly complex process. Many assumptions must be made to develop models that can be subjected to

36

Adedeji, A A USEP, Journal of Research Information in Civil Engineering, Vol. 3, No.1, 2006 and Ejeh, S. P. analysis by the available methods and the models must be verified by experiments.

(a) Design variables

Formulation of an optimum design problem involves transcribing a verbal description of the problem into a well-defined mathematical statement. The formulation process begins by identifying a set of variables to describe the design variables. Once the variables are given numerical values, we have a design of the system. One must be able to change the variables independently to obtain alternative design; in other words, design variables must be independent of each other as far as possible. These variables are regarded as free because the designer can assign any value to them. An important first step in the proper formulation of the problem is to identify design variables for the system.

(b) Design function

The objective function or cost function is a criterion needed to judge whether or not a given design is better than another. To be meaningful, valid objective function is influenced by the design variables.

( c) Design constraints

All systems are designed to perform within a given set of constraints which include limitations on resources, material failure, response of the system, member sizes, etc. The constraints must be influenced by the design variables of the system, because only then can they be imposed; i.e. a meaningful constraint must be a function of at least one design variable.

Most constraints for design problems are expressed as inequalities. For example, the actual member stress must not exceed the material allowable stress.

6. Generic System of Genetic Algorithms

Genetic algorithms are not too hard to program or understand, since they are biological based. Thinking in terms of real life evolution helps to understand the following operation system.

6. 1 Creating a Random Initial State

An initial population is created from a random selection of solution (which is analogous to chromosomes). This is unlike the situation for Symbolic

Artificial Intelligent Systems, where the initial state in a problem is already given instead.

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6.2 Fitness Evaluation

A value for fitness is assigned to each solution (chromosome) depending on how close it actually is to solving the problem (thus arriving to the answer of the desired problem). These “solutions” are not to be confused with

“answers” to the problem, but as possible characteristics that the system would employ in order to reach the answer.

6.3 Selection

According to Dawkins (1986), national selection means differential survival of entities where some entities live and others die. There are many different techniques (such as elitist, linear rank, roulette wheel, stochastic universal sampling and truncated selections) which a Genetic algorithm can use to select the individuals to be copied over into the nest generation. In this work

Tournament selection method (Tang et al, 1996) is employed, where subgroup of individuals is chosen from the large population and the member of each subgroup compete against each other, while only one from each subgroup is chosen to reproduce.

6.4 Reproduction

According to Dawkins (1986), natural selection means differential survival of entities where some entities live and other die. For this to happen there must be a population of entities capable reproduction. In genetic algorithm, the two basic operators for reproduction are: Crossover and Mutation. Performance of genetic algorithm depends on the two aforementioned operations.

6.5 Crossover

Crossover is a Genetic Algorithm operator that entails choosing two individuals to swap segments of their parents. This process is intended to stimulate the analogous process of recombination that occurs to chromosomes during sexual reproduction. In other words, crossover causes genotypes (set of genes that builds a life form or phenotype) to be cut and spliced.

6.6 Mutation

This operator forms a new chromosome by making (usually small) alterations to the values of genes in a copy of a single parent chromosome, just as mutation in things changes one gene to another, so also in a genetic algorithm.

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Binary encoding for single-point crossover was adopted for easy identification of the main variables of short chromosome with less design information for the phenotype.

6.8 Next Generation

If the new generation contains a solution that produces an output that is close enough or equal to the desired answer, then the problem has been solved. If this is not the case, then the new generation will go through the same process ad their parents did. This will continue until a solution is reached. Simple outline of genetic algorithm is shown in Figure 1. And in Appendix I programme procedure for a simple GA pseudo-code is illustrated.

Initial Population

Fitness Objective function

Selection

Crossover

Mutation

GEN > MAX GEN

NO

GEN ~ GEN + 1

YES choose initial population evaluate each individual’s fitness repeat

Stop select individuals to reproduce mate pairs at random apply crossover operator apply mutation operator evaluate individual’s fitness until terminating condition

Fig. 1. Outline of genetic algorithm

7. Genetic Algorithms in Optimal Design

Genetic algorithms were developed to solve unconstrained optimisation problems. However, engineering design problems are usually constrained

(Obayashi et al, 2000). They are solved by transforming the problem to an unconstrained problem. The transformation is not unique and one possibility is to use the following strategy:

Find x = [ b x

1

,…, b x

NEBV

; i x

1

,…, i x

NNIV

; s x

1

,…, s x

NSBV

] (4)

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Minimise : f ( x )

  i c i



Max ( 0 , g i

)

 i

 c j

 h j

( 5 ) where design variables b, i and s = Boolean, inequalities and continuous respectively, c i

and c j

= penalty parameters used with inequality and equality constraints. Determining the appropriate penalty weights is always problematic. An algorithm can be proposed here where the penalty weight is computed automatically and adjusted in an adaptive manner. First the objective function is expressed: f ( x )

 c a i



Max ( 0 , g i

)

 i

 h j

( 6 )

In order to select c a

: (i) c a

is set as the minimum f(x) among all feasible designs in the current generation, if there are feasible designs in the current generation. And if the maximum f(x) among all feasible designs is used, infeasible designs will have a smaller probability to survive even if the constraint violations are small; (ii) c a

is set as the f(x) that has the least constraint violation if there is no feasible design.

The general constrained nonlinear programming problem can be stated as follows:

Minimize f(x) = cost (fitness) function (7a)

Subject to g j

(x)



0, j = 1, 2, …q

(7b) where x = {x

1

, x

2

,….x

n

} T = a vector of n design variables, g j

(x) = constraints functions. This can be transformed into the following model:

Minimise f(x) = cost (fitness) function

Subject to

(8a) g



( x )



1

 ln n  i



1 exp[

 g i

( x ))



0

The parametric constraint evaluation functions

( 8 b )



1

 ln i n 



1 exp[

 g i

( x ))



0 and if

  

the optimisation problems in equations (7) and (8) would have the same Kuhn-Tucker points. In other word equation (8) can be solved by using quasi-exactness penalty function.

40




( x , a )

 f ( x )

 a

 ln i n 



1 exp[

 g i

( x ))



0 ( 9 )

Parameter



can be chosen in the range10 3 to 10 5 and a > 0 is the penalty factor. The fitness function of GA therefore is:

Max F(x, a) = c -



(x, a) (10) c is a large positive number to ensure F > 0.

Typically, cost is the most popular optimisation criterion in structural design.

Cost is a function of the total weight of the structure. Other important factors that may be considered in estimation of the cost of a structure are associated with maintenance (relating to the total surface area for structural element) and connections. For framed structures, where total weight is the only term in the objective function subjected to stress, displacement, and fabrication constraints, the optimisation problem may be defined as:

Minimise, z

W



H  e

 e

L e

A e

( 11 )

Subject to the following constraints:

{

 l

}



{

 all

}



{

 u

}, d l



d all



d u

and A l



A all



A u

(12) where

 e

, L e

and A e

= material density, length and cross-sectional area of the element e respectively, {



}, d and A represent the constraints for stress, displacement and area respectively. Subscripts i and u refer to the prescribed lower and upper boundaries of each constraint.

In GA, to evaluate the performance of a solution string, the string’s binary characters are decoded into values of the design variable representing the properties of the structure. Using these design variables, a structural analysis is performed (using Finite Element method of analysis), after which the value of the objective function as given in equation (12) is computed. Now if any constraints are violated, a penalty (i.e. the degree in which the constraints are violated) is applied to the objective function. Therefore the penalised objective function provides a relatively meaningful measurement of performance of the solution string. Typically the penalty functions are of the form:

41


 i



1

 k









 c c max



1









 n

( 13 ) where

 i

= the penalty constraint i, k = penalty factor, c = value of the constraint from a solution string, c max

= value of the constraint i and n = penalty rate. The penalty objective function of a particular string can be obtained by multiplying the fitness value or cost objective (ie weight of the stricture) by the corresponding penalty factor k. so that:

P



W m

  i

( 14 ) where P = penalized objective function, W = weight of the structural member and m = total number of points where the constraints are checked and the product representing the total penalty for i constraints.

Once the fitness of each string in the population is determined from the penalised objective function values, a new generation of string is produced using reproduction operator (see Reproduction for crossover and mutation (of

0.1%) methods). After a mutation operation is applied to the population a new generation is reproduced.

In this case the population is divided into several partitions. The strings inside each partition are assigned the same probability of being selected as parent strings. This reproduction scheme may be based in the elitist model

(see selection) as proposed by De Jong (1975) where the partitioning may improve local search in the neighborhood of the upper partition of the population. As such global exploration of the search space may be prohibited.

8. Numerical design Example

8.1 Design Information

The stucture: Lecture Theater Building,

Code of Practice: BS 5950

General Loading is shown in Table. 1

8.2 Design Load

Factor of Safety: Live Load = 1.6, Dead load = 1.4

Total Dead Load = 0.515 x 9.12 x 2 = 9.39 KN

Figure 2 shows the plan and section of the Pratt truss system.

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Loading conditions

Table 1. Loadings

Loading ( kN/m 2 )

Roofing Sheet 0.075

Fixing 0.045

Services

Purlins

Self Weight

0.100

0.028

0.267

Live Load 0.250

X

X

20 m

TRUSS SYSTEM PLAN

9 m

1.5m

6 @ 1.5m = 9.00 m

SECTION X-X

Fig. 2 Truss system

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8.3 Imposed and Design Load

Impose loads on roof = 0.250 KN/m 2

Total Impose Load = 0.250 x 9.00 x 2 = 4.50 KN

Design Load (w) = 1.4G

K

+ 1.6Q

K

Load at Node = 3.4 KN

8.5 Design of Steel truss

The roof truss design is grouped into four; the top chord, the bottom chord, the vertical chord, and the diagonal chord. The maximum axial load for each member was used for each section. The tension and compression capacity only were determined here since the purlins was expected to be on the nodes.

No load is expected to be along span of the top members and thus implies that no need for checking the load capacity. Appendix II shows the truss system loadings and dimensions as well as the results of the axial forces used in the design according to BS 5950.

For the construction of the trusses in steel, an Equal angle sections are used.

This section provides the strength and rigidity require of members in such a frame, while at the same time having a good strength-to-weight ratio and easy handling and fixing characteristics. All the designs conform to Table 2 of BS 5950 shows members’ properties.

Table 2. Members’ properties (geometrical and material)

Parameters

Material:

Shape:

Cross-section Area,

Characteristic strength

Allowable compressive stress

Maximum Applied Member loads

Values

Steel

Equal angle

A = 2ht – t 2

p c =

273 N/mm 2

σ a

= 148 N/mm 2

P cmax

= 51.72 KN (Top

Member)

P

T

max

P c max

P

T

max

Applied Stress,

Member Length

L

2

L

3

L

4

51.01 KN (Bottom Member)

10.3 KN (Vertical Member)

14.57 KN (Diagonal Member)

σ = P/A

L

1

= 1.52 m (Top member)

1.50 m (Bottom member)

1.50 m (Vertical member)

2.12 m (Diagonal member)

Source: Adeyemo (2004)

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9. Optimization Procedure

9.1 Problem Formulation

Minimize Z

(h, t)

= 2 ht – t 2

Subjected to:

(a) Stress in Compression where g

1

≡ σ – σ a

≤ 0

σ = P/A

(15)

(16)

(17)

and

2 ht

P c

 t

2



148



0 ( 18 )

(b) Stress in Tension

2 ht

P c

 t 2



275



0 ( 19 )

(c) Other Constraints

P ≤ (2ht – t 2 ) P c

(20)

9.2 Genetic Algorithms Numerical Analysis

To find the Minimum of the function f(h

2

t) = 2ht – t 2 with both h and t as an integer, the code generation involves the use of binary representation of nine bits has been chosen for top and bottom chords where the six bits on the left represent the h value and the three bits on right represent the t value (Table

3). Also, eight bits has been chosen for vertical and diagonal chords where the five bits on right represent the h and three bits on right represent the t value.

Let’s take an optimal steel section of 43 x 43 x 5 i.e. (43, 5). Its cross sectional area A = 405

Then, (2ht – t 2 ) – 405 = 0

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Table 3. Code Generation for top and bottom chords

String h t

Fitness Expected

Count

Actual

100110 101

101010 100

101101 100

100010 110

101000 101

100011 110

38

42

45

34

40

35

5

4

4

6

5

6

0.020

0.012

0.016

0.030

0.033

0.048

0.75

0.45

0.60

1.13

1.24

1.81

1

1

2

1

0

1

For the first string, 2ht – t 2 = 355

Then, 405 – 355 = 50

Therefore, Fitness = inverse of 50 (i.e. 50 -1 ) = 0.020

Fitness Average = 0.0265

0.020

Expected count = = 0.75 ≈ 1

0.0265

By dividing an individual’s fitness value with the average of all fitness value, we can calculate the expected count of this individual in the next generation.

Table 3. Reproduction of Selected String

Selected

String

After

Crossover

& Mutation

h

(mm)

t

(mm)

Reproduction

Actual

100110101 100110110

100011110 100011101

100010110 100000101

101000101 101010110

38

35

32

42

6

5

5

6


Count

- 0.00

0.013

0.009

-

2.24

1.35

0.00

0

2

2

0

101101100 101101110

100011110 101011100

45

43

6

4

-

0.013

0.00

2.24

0

2

After several iterations using the Genetic Algorithm operators Table 4 gives the last iteration in which the result was found.

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Table 4. Final Iteration

String

101010101

101010101

100010101

101010101

100011101

101010101

After

Crossover

h

(mm)

& Mutation

101010101 42

101010101 42

100010101 34

101010101 42

100011110 34

101010101 42

t

(mm)

5

6

6

5

5

5


Count

0.100

0.100

0.011

0.100

0.011

-

1.85

1.85

0.20

1.85

0.20

0.00

Actual

2

0

0

2

2

0

The code generation for vertical and diagonal member uses eight bits of string, where five bits at the left represent h-value and three bits at the right represent the t-value.

Table 5: Code Generation for vertical and diagonal members

String h

(mm)

1011 1101

1011 0100

1010 0110

1100 1101 27 t

(mm)

5

4

4

6

3

5

Fitness

0.06

0.02

0.03

0.06

0.014

-

Expected

Count

1.94

0.65

0.97

1.94

0.45

0.00

Table 6. Reproduction of Selected Strings.

Selected

String

After

Crossover

& Mutation

h

(mm) t

(mm)


Count

Actual

2

0

0

2

1

1

Actual

10111101

10100110

10111101

11001100

10110100

10100110

10110110

10101101

10111100

11001111

10100110

10100100

22

21

23

25

20

20

6

5

4

4

6

4

-

0.03

0.02

0.03

0.06

0.01

0

1.2

1.8

1.2

2.4

0.4

0

1

2

1

2

0

After several iterations, using Genetic Algorithm operators the best solution to the problem was found in Table 7 and Figure 2 gives the graph of fitness against generation.

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Table 7. Final Iteration

Selected

String

After crossover

&

Mutation h

(mm) t

(mm)

Fitness

Expected

Count

Actual

10100110

11000101

10100110

11000101

11000101

11000101

10000101

11100110

10000101

11100110

11000101

11000101

16

28

16

28

24

24

6

5

5

5

6

5

0.01

-

0.01

-

0.20

0.20

0.14

0.00

0.14

0.00

2.86

2.86

0

3

3

0

0

0

Figure 2 shows the graph of fitness against the generation, while the best solution for the vertical and diagonal members was found. in Figure 3.

0.12

0.1

0.08

0.06

0.04

0.02

0

0 2 4 6

GENERATION

8

Fig. 2 Fitness against generation for top and bottom chords

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0.25

0.2

0.15

0.1

0.05

0

0 2 4

GENERATION

6 8

Fig.3 Fitness against generation for vertical and diagonal members

10. Discussion of Results

An important aspect in Genetic Algorithm is to decide which individuals should be chosen as parents for the purpose of procreation. With Genetic

Algorithm, this selection is based on the string fitness: according to the

‘Survival of the Fittest’ principle. Once two parents had been selected the

Genetic Algorithms combines them to create two new offspring. A new string with high fitness appears in the next generation.

Table 8 Comparison of Results for the Truss System

Members Position

Top Member

BS 5950

45 x 45 x 6

Area = 504 mm 2

Genetic Algorithm

42 x 42 x 5

Area = 395 mm 2

Bottom Member

Vertical Member

Diagonal Member

45 x 45 x 6

Area = 504 mm 2

25 x 25 x 5

Area = 225 mm 2

25 x 25 x 5

Area = 225 mm 2

42 x 42 x 5

Area = 395 mm 2

24 x 24 x 5

Area = 215 mm 2

24 x 24 x 5

Area = 215 mm 2

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The results obtained by BS 5950 (1985) was compared with the bit string coded Genetic Algorithm Solution as shown in Table 8. It was observed that the area which is the function of mass obtained by Genetic Algorithm is minimal when compared to that of BS 5950, therefore the best possible solution has emerged.

11. Conclusion

The simple Genetic Algorithm for a problem formulation is a powerful tool that is able to converge rapidly to an optimum value of objective function. In this project, two variable formations were used, but function of more variables was also easily employed.

However, the optimum value of cost function of the material was obtained by minimizing the section of the member, which was later compared to the values obtained from BS 5950. It could be concluded here that the cost function of the GA application is 0.85 of the BS 5950 design application.

Aside the rapid convergence by the GA application, the results obtained are economical and unique and could be relied upon due to its operation simplicity. However in a multi-dimensional problem, the algorithm may require powerful software, such as MATLAB programming facility, for a number of generations following the Pseudo-Code of a simple GA in this work.

1 2 .

.

R e f f e r e n c e s

Adedeji, A. A., (2005), Genetic (Evolutionary) Algorithm: Introduction and

Its Use as an Automatic Structural Design Tool, Seminar Presented to the

Faculty of Engineering, University of Ilorin.

Adeyemo, A. A. (2004). Analysis and Design of Steel Roof Truss of a

Lecture Theatre for 300 seats, A Project report submitted to Civil

Engineering Department, University of Ilorin. pp 1 – 32

Arora, J. S. (1989), Introduction to Optimum Design, McGraw-Hill Inc, pp 1-

12.

British Standards, BS 5950: Structural Use of Steelwork in Building: Part 1:

1985, code of Practice for Design in Simple and Cautious Construction, Part

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Appendix I

5.

6.

7.

8.

9.

10.

11.

12.

13.

Programme Steps of a Simple GA

1.

2.

3.

4.

proc Step_Up(cea) : for s = 1 to MAX_STEPS do

for x = 1 to WIFTH do

for y = 1 to HEIGHTdo

n_list = Compute_Neigh(cea, position(x,y)) ;

Selected_inds = Perform_Selection(n_list) ; aux_pop= Apply_Reproduction_Operators(selected_index) ; end_for ; end-for ;

cea= Replace(cea, aux-pop) ;

Evaluate>population(cea) ;

end_for ; end_proc STEP_UP ;

PseudoCode of a simple GA

Move the pointer over the words

Present a brief explanation with the mean of the main algorithm’s line

LINE 2

The algorithm runs until one of the two following stopping conditions is satisfied:



The algorithm finds an optimal solution to the problem



The algorithm reaches a certain number of generation (represented by

MAX_STEPS) without finding an optimal solution and stops

LINE 3 and 4

In such generation, the algorithm examines all the individuals of the population. This population is hold in a 2_D toroidal grid shape of size WIGTH x HEIGHT (more details).

LINE 5

The neighbours of the individual placed on position (x, y) are stored in an empty list.

LINE 6

52


The parents of the individual of position (x, y) are selected from the list created in line

5 and placed in a neqw list.

LINE 7

The parents of the individual of position (x, y) are selected from the list created in line

5 and placed in a new list

LINE 10

Reproduction

LINE 11 – LINE 13

Evaluate and end

A p p e n d i i x I I I I

Angle at the diagonals

Theta 1 =

Theta 2 =

Theta 3 =

9.46

o Theta 4 =

18.45

0 Theta 5 =

26.57

o Theta 6 =

33.69

o

39.81

0

45.00

o

Length of the Members

BOTTOM TOP

MEMBER

LENGTH (m)

VERTICAL DIAGONAL

MEMBER MEMBER MEMBER

LENGTHS (m) LENGTHS (m) LENGTHS(m)

1.50

1.50

1.50

1.50

1.50

1.50

1.52

1.52

1.52

1.52

1.52

1.52

No. of Nodes and their Loads

NODE NO.

1

2

3

4

5

6

7

0.25

0.50

1.58

1.68

0.75

1.00

1.80

1.95

1.25 2.12

1.50

NODE LOAD (KN)

1.70

3.40

3.40

3.40

3.40

3.40

1.70

53


AXIAL FORCES FOR EACH MEMBER

BOTTOM TOP VERTICAL

MEMBER

FORCE (KN)

MEMBER

FORCE (KN)

MEMBER

FORCE (KN)

DIAGONAL

MEMBER

FORCE (KN)

51.01

40.81

30.08

19.61

9.26

1.04

-51.72

-51.72

-43.02

-32/14

-21.53

-11.03

Span of Roof Truss =

Right Roof Height =

9.00 m

1.50 m

- 3.40

-5.37

-6.98

-8.63

-10/30

-10.20

10.75

12.00

12.58

13.48

14.57

54

Adedeji, A A USEP, Journal of Research Information in... and Ejeh, S. P.

GENERATION

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