optimization of space trusses using genetic algorithm

Zagazig University Faculty of Engineering Structural Engineering Department

OPTIMIZATION OF SPACE TRUSSES USING GENETIC ALGORITHM A Thesis Submitted to the Faculty of Engineering Zagazig University For The Degree of Master of Science in Structural Engineering Presented by

Eng. Osman Hamdy Osman (B. Sc., civil engineering 2009) Supervisors

Prof. Dr. Osman Shallan Head of Structural Engineering Department Zagazig University – Faculty of Engineering.

Prof. Dr. Atef Eraky Bakry

Ass. Prof. Dr. Tharwat Sakr

Prof. of Structural Engineering. Zagazig University – Faculty of Eng.

Ass. Prof., Structural Engineering Dept. Zagazig University – Faculty of Eng.

2015

First of all, I would like to introduce my greatly indebted in my work and success to ALLAH who supported me with patience and strength to complete this thesis. Cordial thanks and deep gratitude are offered to Prof. Dr. Osman Shallan, Head of Structural Engineering Department, Faculty of Engineering, Zagazig University, for his valuable guidance, encouragement and support. I would like to express my thanks and gratitude to Prof. Dr. Atef Eraky Bakry, Professor of Structural Analysis, Faculty of Engineering, Zagazig University, for suggesting the project, and for his great help, guidance, and support. Special thanks are offered to Associate Prof. Dr. Tharwat Sakr, Associate Professor, Structural Analysis, Faculty of Engineering, Zagazig University, for his guidance, support, patient and stimulating directions throughout the work also for the time he spent in refining the manuscripts and discussing the subject with me.

Osman Hamdy Osman

I

ABSTRACT Optimization of truss structures is a popular topic in structural and mechanical engineering due to their benefits to industry through the reduction of structure weight which leads to low costs and environmental friendly buildings. So optimization is important part of the design process. Size, shape and topology optimization are known categories of structural optimization aim at the selection of joint coordinates, member connectivity and element cross sections. Performing optimization for these three categories simultaneously is considered a challenge due to complexity and the restriction on the number of available cross sections used as variables. In this study, an approach for weight minimization of trusses has been developed using genetic algorithm. The proposed approach changes the design variables which constitute the Genetic Algorithm (GA) chromosome. Instead of using the nodes coordinates, member connectivity and member cross sections as design variables, it uses only free nodes coordinates and non-support nodes deflections. Topology and sizing optimization can be included using mechanical relationships based on the assumed deflections. The proposed approach was incorporated into MATLAB code to perform size, shape and

II

topology optimization simultaneously avoiding the complexity of traditional technique. Also deflection was used as design variable to avoid the restriction on available used cross sections number. Also making topology optimization using simple condition depends on the elongation of each member. Three benchmark problems were explored: size, shape, topology optimization for a10-bar plane truss, 25-bar space truss and size optimization for 72-bar space truss. The presented results produce better optimum weight than previous works in literature using less computational costs.

III

Table Of Contents ACKNOWLEDGMENTS …………………………………………………………..…………… I ABSTRACT ………………………………………………………………..…………..……..…. II TABLE OF CONTENTS ………………………………………………….……………....…….. IV LIST OF TABLES ……………………………………………………………………………….. VII LIST OF FIGURES …………………………………………………………………….….…… VIII

CHAPTER (1)

INTRODUCTION

1.1- General …………..…………………………………………………..………………..… 1 1.2- Motivation of the Study…………………………………………….……….…….……..

3

1.3- Research Objectives…………………………………………………….………….……

4

1.4- Layout of Thesis……………………………………………………………….…….…..

5

CHAPTER (2)

BACKGROUND AND LITERATURE REVIEW

2.1- General ….. ……………………………………………………………………….…….. 7 2.2- Optimization of Trusses…………………………………………………………….……. 8 2.3- Evolutionary Techniques ………………………………….…………………….………. 9 2.4- Genetic Algorithm ………………………………………………………………...……. 10 2.4.1- The GA Principle……………………………………………..…………….……. 11 2.4.1.1- Representation………………………………………………..……….. 13 2.4.1.2- Fitness Evaluation………………………………………….…………. 14 2.4.1.3- Selection………………………………………………………....……. 14 2.4.1.4- Cross-Over………………………………………………………..…… 15 2.4.1.5- Mutation………………………………………………….…......……. 16 2.4.1.6- Population Size………………………………………………..………. 17 IV

Table Of Contents 2.5- Structural Engineering Applications…………………...………………………………. 17 2.5.1- For Truss Structures………………………………………………….…...….…. 17 2.5.2- For Concrete Structures……………………….…………………....….….….…. 22 2.5.3- For Composite Structures……………………….……………….…….……..…. 23 2.5.4- Damage Detection………………………………………………………..…..…. 24

CHAPTER (3)

THE PROPOSED APPROACH

3.1- General …..…………………………………………..….………………..….………… 25 3.2- Design Variables in Traditional Technique…………………………………….……… 25 3.3- Traditional Variables Drawbacks……………………...……………………….……… 29 3.4- The proposed approach……………………………………………………..….….....… 32 3.4.1- Proposed Design Variables………………………………………………...….… 32 3.4.2- Extension to Sizing and Topology………………………………….…..…..…… 34 3.4.3- Proposed versus Traditional approaches………………….……….…..…..…….. 39 3.4.4- Handling of Constraints……………………………………………………....…....41 3.4.4.1- Constraint 1: Stability………………………………………..……....… 42 3.4.4.2- Constraint 2: Constructability………………………………....….…… 42 3.4.4.3- Constraint 3: Member stresses………………………………..…..…… 43 3.4.4.4- Constraint 4: Nodal displacements…………………………….…….… 43 3.5- The Proposed Algorithm………………………………………………………….….… 44

V

Table Of Contents CHAPTER (4)

CASE STUDY AND RESULTS

4.1- General …..……………………………………………………………………..…… 48 4.2- Benchmark Ten Bar Plane Truss ……………………………………………….…… 48 4.3- Benchmark 25-Bar Space Truss ……………………………………………..……… 58 4.3.1- Sizing Optimization of the 25-Bar Space Truss………………………..…... 58 4.3.2- Sizing, Shape and Topology Optimization of the 25-Bar Space Truss………68 4.4- Benchmark 72-Bar Space Truss ……………………………………..……………… 76

CHAPTER (5)

SUMMARY AND CONCLUSIONS

5.1- Summary …………………………….…………………………….….….….……… 92 5.2- Conclusions……………………………………………………………..…………… 92 5.3- Suggestions for Future Work………………………………………….….……….… 94

REFERENCES…………………………………………………………….………… 95

VI

List Of Tables Table Title Page (3.1) Comparison between number of bites in traditional and proposed approach………….………………………..….. 34 (4.1) Optimization results for 10-bar plan truss.……….………. 53 (4.2)

Results of previous works with same conditions..……...…

57

(4.3)

Coordinates of the joints of the 25-bar space truss...………

60

(4.4)

Group membership for 25-bar space truss....………..……

60

(4.5)

Loading conditions for 25-bar space truss……….……….

60

(4.6)

Member stresses for 25-bar space truss.……….………….

64

(4.7)

Sizing optimized 25-bar space truss results……………….

67

(4.8)

Coordinates of coordinately optimized nodes of 25-bar space truss.…………………………………………..……. Member stresses of optimized 25-bar space truss.……...…

71 72

(4.10) Size, shape and topology optimized 25-bar space truss results……………………………………………………… (4.11) Members grouping for 72-bar space truss……….……...…

75 78

(4.12) Load cases for 72-bar space truss……………….…………

78

(4.13) Members stresses for 72-bar space truss…………………..

89

(4.14) Optimization results for 72-bar space truss………………..

91

(4.9)

VII

List Of Figures Fig.

Title

(2.1)

A truss structure with its corresponding theoretical

Page

model……………………………………………………

8

(2.2)

Examples of truss optimization categories………………

9

(2.3)

Flowchart of simple GA …………………………..……

12

(2.4)

Illustration of Pareto set for a bi-objective optimization problem ………………………….…..………….………

(3.1)

10-bar plane benchmark truss.…………….……..…....…

20 28

(3.2)

Chromosome sample for 10-bar plane truss. ……………

35

(3.3)

The master flow chart of the proposed.. ……………..…

47

(4.1)

Structure of benchmark ten bar plane truss....………...…

49

(4.2)

Convergence history of benchmark ten bar plane truss structure…………………………………………………

(4.3)

51

Optimized structure of the benchmark ten bar truss for current…………………………………………………..

52

(4.4)

Load case of the ten bar truss in SAP Program (N)……..

54

(4.5)

Maximum deflection of the ten bar truss in SAP Program (mm).…………………………………………..

54

(4.6)

Optimized structure of the 10-bar truss Deb and Gulati....

55

(4.7)

Benchmark 25-bar space truss structure............................

59

(4.8)

25-bar space truss structure model in MATLAB..............

62

(4.9)

Convergence history of 25-bar space truss……...............

63

(4.10)

Load case of the 25-bar space truss in SAP program…….

65

(4.11)

Maximum deflection of the 25-bar space truss in SAP program ………………………………………………….

(4.12)

66

Shape, sizing and topology optimized 25-bar space truss structure model in MATLAB………………………...….. VIII

69

Fig.

Title

Page

(4.13)

Convergence history of 25-bar Space truss structure….…..

70

(4.14)

Load case of the 25-bar space truss in SAP program .…….

73

(4.15)

Maximum deflection of the 25-bar space truss in SAP program……………………………………………………. 73

(4.16)

Benchmark 72-bar truss structure (a) profile view, (b) element and node numbering system…………………….... 77

(4.17)

Convergence history of 72-bar space truss structure………. 80

(4.18)

Benchmark 72-bar truss structure Model in MATLAB Program.………………………………………………...…. 81

(4.19)

Load case 1 for 72-bar truss structure in ETABS program... 83

(4.20)

Maximum deflection of load case 1 for 72-bar truss structure in ETABS program………………………………. 84

(4.21)

Load case 2 for 72-bar truss structure in ETABS program... 86

(4.22)

Maximum deflection of load case 2 for 72-bar truss structure in ETABS program …………………………..….

(4.23)

87

Axial load in members due to Load Case 2 for 72-bar truss structure in ETABS program………………………………. 88

IX

Chapter (1)

Introduction

Chapter (1) INTRODUCTION 1.1 General Performing engineering design optimization, especially the optimization of complex structural systems, is an active area of research. It has become a valuable tool for engineers and designers in recent years. Although it has been applied for over fifty years, optimization in engineering has not been a commonly used design tool until high performance computing systems were made widely available. Structures are becoming lighter, stronger, and cheaper as industry adopts higher forms of optimization. This type of problem solving and product improvement is now a crucial part of the design process in today’s engineering industry. The main objective of structural optimization is saving material taking in consideration the design constrains such as allowable stresses and deflections, In other words structural optimization aims to achieve proper and optimal use of material. Saving material means saving cost and money besides also time consumed in construction process which is considered the most important factor needed to be saved in any project. Truss optimization can be classified to three categories: topology, size, and shape. All three categories generally have the 1

Chapter (1)

Introduction

objective of weight minimization without exceeding stress and displacement constraints. Topology optimization is concerned primarily with the placement of members in the truss structure relative to each other. It also deals with the number of members and the number of nodes that exist in the structure along with their support conditions [1]. Sizing optimization is concerned with the optimal selection of the cross–sectional areas of the truss members where all the nodal connectivity and the nodal locations remain fixed in the truss [1]. Shape optimization is concerned with the optimal selection of the location of the nodal points, which affect the lengths of individual members and the overall shape of the truss, which primarily refers to changing the nodal co-ordinates during optimization [1]. Genetic Algorithm (GA) is the oldest and most common form of evolutionary computation. It derives its behavior from natural evolution and genetics, following Darwin’s major principles of evolution. This method relies on random actions, and survival of the fittest to evolve solutions to optimization problems. It acquires its strength from the fact that a wide variety of problems can be driven to very good solutions by recombining parts of previous good solutions. As engineers and designers search for new optimization methods, they find that the GA can produce results never possible before.

2

Chapter (1)

Introduction

1.2 Motivation of the Study Complex structures become difficult to optimize when variable interactions increase especially when shape, size and topology optimization are made at the same time. Classical GA optimization method can produce sub-optimal results because of these interactions besides the long time consumed due to long chromosomes which depends on the number of variables and the variables space. This leads to huge number of possibilities. This may suggest that this kind of optimization should be carried out using parallel computing where the work load is divided to a group of processors. With limited computer power, the number of possibilities should be kept at a corresponding level; otherwise the calculation time will be extreme. This research is motivated by the need to reduce time consumed and complexity of making many optimization categories at the same time. GA has been shown to provide global exploration capabilities and have been used for design optimization. Many researchers worked on problems but they faced challenge in truss problems due to the existence of three categories of optimization which produce long chromosomes to describe these three categories. This leads to long execution time and complexity which direct some researchers to make optimization in two stages to overcome the complexity. Other researchers made optimization for one or two categories in the same time leading to suboptimal solution due to neglecting the third category. On the other hand the chromosome length depends on the 3

Chapter (1)

Introduction

variable limits so that some researchers tried to use low range for variables to reduce the chromosome length. For example, most of researches used low number of available cross sections to make sizing optimization.

1.3 Research Objectives The objective of this research is to develop a new approach to overcome the pre-mentioned drawbacks by using non-traditional variables. To achieve this goal new optimization strategy using GA has been developed here. It depends on using nodes coordinates and deflections as variables to make size, shape, and topology in the same time. This new technique uses only nodes coordinates and deflections as variables instead of the three traditional variables. Beside the limit of deflection variable is less than traditional cross section variable. So the chromosome which is needed in new approach is highly shorter than traditional one. And that leads to reduce the computational effort, complexity and time consumed for solution. Using deflections as design variables instead of member cross sections can lead to narrow limits of design variables and allow using any number of cross section without affecting the problem size. Also deflection is associated with nodes and cross section is associated with members. Usually the number of nodes is less than the number of members in the truss. This difference between number of members 4

Chapter (1)

Introduction

and nodes becomes high in space trusses. So using deflection as variable instead of cross section is an effective technique especially in complicated space truss where one node can connect three or four members or even more. On the other hand by looking for topology optimization (connectivity optimization), each member has two probabilities which are exist or not exist in the proposed truss. Topology optimization in this suggested strategy is not represented in the chromosome. It is derived by using a simple condition that depends on the elongation of each member which is calculated from deflections. Thus, sizing and topology

optimization

parameters

are

not

included

in

the

chromosome. They are replaced by the node deflections leading to shorter chromosome to overcome the complexity of making simultaneous sizing, topology and shape optimization.

1.4 Layout of Thesis The layout of the presented thesis is arranged as follows: Chapter one is an introduction showing the motivation of this study and its objectives. Chapter two gives a background of the structural optimization and GA method together with the required review of literature. At first, structural optimization is defined and its importance is emphasized. Literature about the optimization of trusses and GA as an evolutionary technique is then presented. Finally, different 5

Chapter (1)

Introduction

types of structural engineering applications are demonstrated. Chapter three presents the proposed technique of truss optimization using GA. It starts with design variables in traditional technique and their drawbacks followed by illustration of the proposed approach and its steps and the master flow chart of the proposed approach. Chapter four shows numerical applications of the proposed approach on three benchmark truss problems. There are 10-bar plane truss, 25-bar space truss and 72-bar space truss. It also contains comparing the results with the previous works in literature. Chapter five contains the summary of the thesis, main conclusions and the suggestion for future works in the thesis fields of interest.

6

Chapter (2)

Background and Literature

Chapter (2) BACKGROUND AND LITERATURE 2.1

General A structure in mechanics is defined by Gordon [2] as “any

assemblage of materials which is intended to sustain loads.” Optimization means making things the best. And the most proper definition of structural optimization field from economical viewpoint is making the structure weight as light as possible. The reduction of the weight of the structure obviously reduces the cost of material used leading to less total costs. Simultaneously, sustainability measures recommend the use of less material with lower embodied energy and carbon dioxide emulsion which lead to green, leadership in energy and environmental design LEED certified structures. Those points give the field of structural optimization a very high importance which necessitates being a part of design procedures. Constraints play an important role in structural optimizations such that it cannot be performed without the consideration of design constraints such as stresses, deflections and stability. Decreasing the cross sections is expected to push the structure to critical limits of stresses and deformation leading to unsafe or even unstable one. Approximately

all

engineering

optimization

problems

contain

constraints of code requirements, stress limits, serviceability conditions

7

Chapter (2)


and structural rules incorporated in the optimization process using too many techniques [3].

2.2

Optimization of Trusses Truss structure is a collection of nodes (vertices) that are

connected by a set of elements (edges) often arranged in a triangular shape. Theoretically, the bars in a truss are assumed to be connected to each other by friction-free joints. In real-life trusses though, the joints are more or less stiff due to welding or screwing the bars together. Even with some stiffness in the connections, a model with friction-free joints can accurately be used if the center of gravity axis of each bar meets in the point where the joint is exist in the model [4] as shown in figure (2.1):

Figure 2.1: A truss structure with its corresponding theoretical model [3].

Figure (2.2) shows examples of the three types of truss optimization. The topology example shows changes in connectivity between several internal elements while the node locations remain constant. The shape optimization shows a constant topology with a variation in node locations. The size optimization shows several examples of frame element cross sectional geometries that might be

8

Chapter (2)


applied to an element. Note that truss elements may only vary their cross sectional area due to the fact that this is the only property that is needed to describe the element. Topology (connectivity)

Shape

Sizing

Figure 2.2: Examples of truss optimization categories [4]

2.3 Evolutionary Techniques Particle swarm optimization (PSO) is a population-based stochastic approach for solving continuous and discrete optimization problems developed by Eberhart and Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling [6]. Ant Colony Optimization (ACO) is a probabilistic technique for solving computational problems introduced by Marco Dorigo in 1992, inspired by the behavior of real ant colonies [7]. Big Bang - Big Crunch optimization (BB-BC) is a heuristic population-based

evolutionary

optimization

9

method

introduced

Chapter (2)


by Osman and Eksin in 2006, inspired by the theories of the evolution of the universe [8]. Huang and Liu [6] carried out size optimization for 10- bar plane truss using particle swarm (PS). Camp [8] performed size optimization for 72-bar space truss using Big Bang – Crunch Algorithm. Li et al. [9] carried out size optimization for same truss using Heuristic Particle Swarm Optimization. Kaveh and Shojaee [7] optimized the size for the same truss using Ant Colony Optimization.

2.4 Genetic Algorithm John Holland is commonly known as the father of the GA technique. He consolidated the technique in his book Adaptation of Natural and Artificial Systems in 1975 [10, 11, and 12]. At this time though, the idea of mimicking the evolution in programming had been around for a while. In Germany, for instance, Rechenberg and Schwefel developed the evolutions strategies in the 1960s. At the same time, similar work was conducted in the USA under the name Genetic Programming. These early proposals involved mutation and selection, but not recombination, which is the key feature of GAs. Even though this new technique gave some promising results, it didn’t gain much interest at the time, probably due to the lack of computational power [12]. Over the next decade the number of scientific publications on GAs grew at approximately 40 % each year till 1995 when it peaked [13].

10

Chapter (2)


The main part of these publications was different implementations of GAs. When it comes to structural optimization, Goldberg seems to be the first one to suggest the use of GAs [14, 15, and 16] in engineering design. In 1986, he used the GA technique to minimize the weight of a ten- bar aluminum plane truss and 25-bar aluminum space truss [11]. These structures are commonly used as benchmark problems in structural optimization.

2.4.1 The GA principle GAs have three characteristic operators namely, selection, crossover and mutation. In each iteration, or generation, these operators are applied on a population of possible solutions, or individuals in order to improve their fitness. Each individual is represented by a string and these strings remind very much of the natural chromosomes, leading to the name of GAs [10]. Initially, the population is created randomly, and the breeding continues until a stopping criterion is reached, e.g. the exceeding of a certain number of generations, or the absence of further improvements among the individuals. In the following sections, a more detailed review of the different GA operators is given. There are many advantages with the GA technique, primarily its simplicity and broad applicability. It can easily be modified to work on a wide range of problems [17], as contrary to

traditional

search

methods that are specified on a certain type of problem [10]. The technique is relatively robust as well; it does not tend to get stuck in

11

Chapter (2)


local optimums as other techniques may do [10, 17]. Furthermore, due to the use of function evaluations rather than derivatives, it can handle discrete variables and is able to work in highly complex search spaces [17]. Figure (2.3) shows the classical flowchart for simple GA where it starts with evaluating fitness of initial population followed by the main three characteristic operators selection, cross over and mutations to produce the next generation.

Initial Population Fitness Evaluation Selection Crossover No

Mutation

Criteria Satisfied? Yes

Final Population

Figure 2.3: Flowchart of simple GA [18]

12

Chapter (2)


2.4.1.1 Representation Just like the chromosome, GA string has different segments, or genes where the data are stored which are called bites and GA is only able to deal with these bites. So the variables limits numerical values should be translated to bites. That translating process called coding process. So every solution values suggested by GA will be in coded format. After that decoding process begins to translate those coded values into decimal values to calculate the fitness of suggested solutions. In the traditional GA, the string consists of a fixed-length binary string [17]. The number of genes is dependent on the number of variables that needs representation. For instance, the following string consists of five genes, g1-g5, representing five design variables:

The number of bits required in a certain gene for every design variable is calculated as [10]: 𝑁𝑜. 𝑜𝑓 𝐵𝑖𝑡𝑒𝑠 = 𝑙𝑜𝑔2 [

ximax −ximin ∆i

]

(2.1)

where ximin is the lower bound of variable i, ximax is the upper bound of variable i and ∆i is the desired precision in variable i.

13

Chapter (2)


For example if the variable X has lower limit 0 and upper limit 10 with desired precision 1 so the number of bites needed to express this variable are 4 bites and the coded lower limits value will be 0000 which represents 0 where upper lower limit is 1111 which represent 10 and any binary coded string between these two strings will represent value between 0 and 10

2.4.1.2 Fitness evaluation Fitness reflects the performance of each individual so it is the measure which affects the selection of the qualified fit individuals for the next generation. In engineering optimization the fitness usually represents the objective function to be maximized or minimized and function is usually expressing the most valuable factor in the problem. For example in structure optimization problem, objective function represents the weight of structure and the goal is minimizing that weight, taking in consideration safety constrain. On the other hand if the problem is about getting the most optimized track between two points fitness will expresses the length of track. So as the track is shorter as the solution is more optimized, taking in consideration some constrains.

2.4.1.3 Selection For the reproduction, individuals with good fitness are chosen to form a mating pool. There exist many different ways to choose individuals, but the main idea is that the better the fitness is, the higher

14

Chapter (2)


the probability to choose [10, 12, 17]. The mating pool has the same size as the population, but good individuals are more frequent due to duplication. A popular selection method is the tournament selection [10, 19]. In this method, small “tournaments” between randomly selected individuals are held, simply meaning that the individual with the best fitness in the group is selected. With a population size of N, N tournaments are held to fill the mating pool. This way, no copy of the worst individual is selected [19].

2.4.1.4 Cross-over With the hope of finding better solutions, the strings in the mating pool are crossed over with each other with the intention of creating a better population. Just as in the selection, there are different cross-over operators, but the main idea is that two random individuals from the mating pool are chosen as parents, and some portion of their strings are switched to create two children [10]. Three usual cross-over methods are given below [10, 17]: Single point cross-over: The two parent strings are cut at a random spot, and the pieces are put together to make two children:

Two point cross-over: The parent strings are cut twice at two random spots to create the children:

15

Chapter (2)


Uniform cross-over: Each bite in the child strings are copied from either one of the parents in random process at a 50 % probability:

2.4.1.5 Mutation In the creation of new chromosome, there is always a small probability for each bit in the string to change from 0 to 1 or vice versa. If so, the child is mutated:

The purpose of this feature is to maintain the diversity amongst the individuals [10], and to prevent the algorithm from getting stuck in a local minimum [17]. The mutation probability should not be too high since in that case the GA turns into random search [17].

16

Chapter (2)


2.4.1.6 Population size The size of the population should be chosen according to the complexity of the problem. In highly complex problems, the gene pool needs to be extensive enough so that the whole search space can be explored [17]. But of course, with a bigger population the computational time and effort is increased, so the upper limit of the population size should be determined by available computer power and time.

2.5 Structural Engineering applications 2.5.1 Truss Structures Rajeev and Krishnamurthy [20] and Jenkins [21] were among the first to apply GA successfully to truss sizing optimization taking in consideration stress and deflection constrains.

They applied sizing

optimization for 10-bar benchmark plane truss and for 25-bar benchmark space truss. Several researchers' in the mid-1990s tried to work on the limited topology and sizing optimization from different prospective. Hajela and Lee [22] performed topology and sizing optimization for 10 bar benchmark plane truss and for simple bridge truss type; they followed a two stage procedure. In the first stage topology optimization has been carried out for the ground structure where each nodes connected to each other nodes i.e. all possible members exist taking in consideration

17

Chapter (2)


stability constrain. Many proposed topology optimized solutions for that truss are gotten from first stage. The next stage used the optimized topologies generated in the first stage as initial seeds and member-sizing optimization was performed with the consideration of stress and deflection constraints in addition to the ability of adding or removing some members in the second stage without affecting the stability of the truss. Rajan [23] performed sizing and shape optimization only and sizing with shape optimization (considering support coordinates are variable) along with limited topology optimization and taking into consideration structural constrains for 10-bar benchmark plane truss and 14-nodes plane truss. For limited topology optimization, Rajan suggested that there are some essential member for each certain problem and must exist while topology optimization is performed for non-essential members using Boolean variable 0 and 1 values where 0 means the member absence and 1 means member presence. Rajeev

and

Krishnamoorthy

[24]

presented

a

GA-based

methodology for optimal design which simultaneously considers topology, configuration, and cross-sectional parameters in a unified manner. They applied this manner on 10-bar benchmark plane truss and 18-bar plane truss. Their methodology is a two-phased approach and can handle both discrete and continuous design variables. In this procedure, they used GA to obtain appropriate lower-bound indices for each design variable in Phase 1. In Phase 2, these indices were then

18

Chapter (2)


improved in an adaptive manner, in order to achieve the optimal solution. The method used a variable string length GA, which allowed variations in topology, size, and configuration. Cheng and Li [25] presented a methodology of constrained multi objective optimization problems (MOP) by integrating a Pareto GA and a fuzzy penalty function method. The Pareto GA proposed consists of the Pareto set filter and the niche as two additional operators in addition to the reproduction, crossover, and mutation to constitute five parts system. The Pareto set filter was applied to pool non-dominated points ranked 1 at each generation and drop dominated points in order to stop the loss of Pareto optimal points. Ranking was applied as a continuous labeling process such that at each generation non-dominated points (Pareto set) are selected and assigned rank 1. From the remaining population non dominated points are identified and assigned rank 2 and this process continues for rank 3, 4 and so on until the entire population is ranked. A Pareto optimal solution cannot be improved with respect to any objective without worsening at least one other objective. Points closer to Pareto optimal set is assigned higher fitness values. Any point in a pare to optimal set can become an "optimum solution" depending on the decision-makers opinion as MOP has no unique optimum that can simultaneously optimize all objectives as shown in figure (2.4).

19

Chapter (2)


Figure 2.4: Illustration of Pareto set for a bi-objective optimization problem.

Niche technique prevents genetic drift and significantly distributes a population uniformly along a Pareto optimal set. In such technique an offspring replaces its parent if the offspring's fitness exceeds that of the inferior parents. This niche technique prevents the formation of a lethal and the reproduction of procedure is a steady state one. The revised penalty function method was reported to fail to work properly in a Pareto GA for constrained MOP. Thus, a fuzzy logic penalty function method is used. GA with crossover probability of 60 percent and mutation probability of 1 percent was applied. Sample cases of 4- bar pyramid truss, 72-bar space truss with two criteria (Minimum Weight and Minimum Strain energy) and four-bar truss with two criteria (Minimum weight and Minimum Deflection of two load cases). Further improvement in the field was shown by Deb and Gulati [26], in which each member’s presence or absence in the truss topology was determined by the area assigned to each member of a truss. If the

20

Chapter (2)


member size was less than the predefined critical area, then the member was removed. In their research, the nodes of the trusses were divided into two categories. The first category was known as the basic nodes and then extras non-basic nodes. The presence of the basic nodes was a must to make the trusses kinematically stable. The non-basic nodes were optional. Thus, the trusses that did not have the basic nodes were excluded from further evaluation. This simultaneously size and topology optimization method was applied on 10-bar benchmark plane truss [26]. Jenkins [27] developed an idea of implementing GA without crossover. He proposed an adaptive GA that used only mutation. In his research, two kinds of mutation were used. One was the random mutation, which was similar to that used in simplified GA SGAs, in which mutation is performed with genes that are selected based on mutation probability, and the other was intelligent mutation that was performed conditionally. For instance, if the stress in a truss member exceeds the design stress then positive mutation was applied to increase the cross sectional area otherwise negative mutation is performed. Thus, this type of implementation reduced the computational expense of the method. Hultman [3] carried out size, shape and topology optimization for 10-bar benchmark plane truss using steel material while previous works used aluminum. He took into consideration allowable stress, deflection and buckling according to

Eurocode. Topology

optimization has been carried out using sizing chromosome by

21

Chapter (2)


adding 50% of available sections number as zero section i.e. its cross section area is zero and whenever the members get zero sections this member will be removed so the probability to remove the member is 30%. Two runs have been carried out. The first run produced optimized size, shape and topology truss but there are some indications that the topology is suboptimal so another run has performed for the optimized one which is gotten from the first run.

2.5.2 Concrete Structures Rafiq and Southcombe [28] introduced an approach using GA to optimal design and detailing of reinforced concrete bi-axial columns. The used design variables were optimal bar size and bar detailing. The procedure attempted to keep the sectional moment capacity and cross section and get the minimum reinforcement area

leading

to

more

economical

design.

The capacity

calculation and reinforcement arrangement are made according to the British Standard (BS8110). Coello et al [29] presented a GA optimization model for the design of rectangular reinforced concrete beams subjected to a specified set of constraints. The model considered the minimization of the cost of the beam considering the costs of concrete, steel and shuttering. Simple GA was applied and results were compared to those obtained via geometric programming. Strength design procedure was adopted for which the ultimate concrete strain is 0.003 and trapezoidal stress distribution was assumed.

22

Chapter (2)


The optimization of construction costs of mass concrete structures such as dams, foundation slabs, and bridge decks was carried out by Fairbairn et al. [30]. The proposed model used the GA in selecting the material type, placing temperature, height of lifts and time interval between lifts as design variables. These variables control the cost of mass concrete structures which was used as the objective function. Peng and airfield [31] presented an integrated design optimization combining the mechanism method with GAs for the optimization and design of arch bridges.

2.5.3 Composite Structures In their review of methods used to optimize composite panels, Venkataraman and Haftka [32] reported that GAs have been the most popular method for overcoming the optimization complexity of composite panels. This can be attributed to the fact that the problem is discrete in nature such that the ply orientation must be one of specific values produced by the manufacturer. Okumura et al [33] carried out Optimum design for the weaving

structure

of

three-dimensional

(3-D)

reinforced

composites. Soremekun et al [34] explored several generalized new procedures for the design of composite laminates. The problem design variables are discrete as ply angles and ply thickness can only be available in manufacturer specific values. Maximizing the buckling load of simply supported composite plate and maximizing

23

Chapter (2)


the twisting displacement of cantilever composite plate was set as the two objectives of the analysis.

2.5.4 Damage detection Rao et al. [35] have proposed a method for locating and quantifying the damage in structural members using the concept of residual forces. GA has been applied for the minimization of objective function consisting of the sum squared diagonal terms of the residual force matrix. Friswell et al. [36] applied GA to eign-sensitivity analysis of damage in structures. The objective of the study was to identify the position of one or more damage sites in a structure and to estimate the extent of damage at these sites. Design variables include discrete values of damage location and continuous variable indicating the extent of damage as a percentage reduction in stiffness.

24

Chapter (3)

The Proposed Approach

Chapter (3) THE PROPOSED APPROACH 3.1 General Complex structures become difficult to optimize when variable interactions increase especially when shape, size and topology optimization are made at the same time. In this chapter, the proposed approach for GA optimization of trusses is discussed. Design variables are first introduced showing the difference between traditional variables used for combined shape, topology and size optimization and current proposed design variables. By applying the traditional design variables to applicable example, their drawbacks are illustrated. Then the proposed approach and its variables are introduced and applied to the same example illustrating the advantages of proposed approach over traditional method. Constrains are illustrated and the master algorithm and its flow chart are explained.

3.2 Design Variables in Traditional Technique In truss optimization problems, the traditional optimization variables used to represent the shape, size and topology can be listed as follows: 1-

Nodal coordinates for shape optimization containing the x, y and z coordinates of all nodes of the truss depending on the number of nodes and whether the truss is plane or space one. 25

Chapter (3)

2-


Members cross sections for sizing optimization containing the cross section area "A" for each member of the truss depending on the number of members in the truss and the available sections for member selection.

3-

Topology matrix showing members presence for topology optimization. It contains two options one option represents the presence and other represents the absence for each member of the truss depending on the number of members in the truss. In GA, all optimization problem data are stored in the GA

chromosome leading to a chromosome length as long as the data stored in it. More data lead to more chromosome length and consequently to more time consumed and more proposed solutions for the problem. The chromosome length always depends on three parts: A- The number of optimization categories or variables which need to be optimized for the truss from the three optimization categories shape, size and topology optimization such that. Nc

L = ∑ Lc

(3.1)

c=1

where L is the total length of chromosome;

Lc

is

the

length of chromosome for category c; and Nc is the number of optimization categories applied topology and sizing, etc.).

26

(1: for sizing only, 2: for

Chapter (3)


B- The number of objects subjected to each optimization category or variable which will be optimized, i.e. the number of members for size and topology optimization and the number of nodes for shape optimization as following equation. Lc = Nc ∗ Tc where Nc

(3.2) is the Number of objects applicable to the

optimization category (No. of members for size optimization, No. of nodes for shape optimization, etc…); and Tc is the length of chromosome per object (Number of Bits). C- The number of available profiles for each objects in the optimization category i.e. the number of available section for Size optimization or the upper, lower limits and the precision of coordinates for shape optimization as presented by following equation. ximax − ximin Tc = log 2 [ ] ∆i where ximin

(3.3)

is the lower bound of variable i; ximax

is the

upper bound of variable i; and ∆i is the desired precision in variable i. The design variable for the traditional approach is then composed of two or three coordinates for each node, cross sections and absence/presence case for each member of the whole truss. As an example, consider the 10 bar plane truss shown in figure (3.1) where nodes number 5 and 6 are coordinately variable nodes 27

Chapter (3)


which are exposed to shape optimization while other load and support nodes are fixed and all members are exposed to sizing and topology optimization.

Figure 3.1: 10-bar plane benchmark truss.

The chromosome will consist of three parts: A-

For Shape Let the desired precision for node coordinates be 0.1 m. From truss geometric limits min. X-coordinate is 0.0 m and max. Xcoordinate is 18.28 m while min. Y-coordinate is 0.0 m and max. Y-coordinate is 9.14 m. Number of bites for X-coordinates of one node = log 2 [

18.28−0

]

0.1

= 7.51 = 8 bites. Number of bites for Y-coordinates of one node = log 2 [ 6.51 = 7 bites. 28

9.14−0 0.1

]=

Chapter (3)

B-


For Sizing The desired precision is one section. Total number of available sections here is 226 sections used by Hultman [3]. Number of bites for size variable of one member = log 2 [

226−0 1

]

= 7.82 = 8 bites. C-

For Topology Total number of available probability is two (presence/absence) Number of bites for topology of one member = log 2 [

2−0 1

]= 1

bite. Combining Chromosome Total number of Chromosome bites = (number of bites for X coordinates + number of bites for Y-coordinates) × number of variable nodes + (No. of bites for sizing + number of bites for Topology) × Total number of truss members. Total number of Chromosome bites= (7+8)× 2+ (8+1)× 10 =120 bites.

3.3 Traditional Variables Drawbacks Traditional cross section variable depends on selecting randomly the proposed cross sections for each member from the available cross sections. The fitness value considering constrains 29

Chapter (3)


penalty are calculated for the whole proposed truss as following equation [15, 3, 20, 7]. F(x) =

1

(3.4)

f(x)(1000v+1)

where F(x) is fitness value, 𝑓(𝑥 ) weight of truss and 𝑣 is the count of the number of constraints violated by a given solution including check of stability and constructability. So some times the individual chromosome of truss produces optimum topology, shape and member size. But due to random selection of cross sections, this combination produces section(s) smaller than minimum needed cross section(s) and this causes exceeding constrain(s) limit(s) such as stress or deflection. This unfit selected section(s) leads to unfit structure due to the penalty resulting from design violations. If these cross section(s) are increased slightly until reaching the minimum safe cross section(s), the truss will be valid and has good fitness, i.e. unsafe stress of only one member or unsafe deflection of only one node is enough to consider the whole truss unsafe. It is because of the penalty resulting from equation (3.4). In some cases individual chromosomes combination produces a good topology, shape and member size but due to random selection this combination produces section(s) larger than minimum needed cross section(s). So the outcome truss is overly weighted despite if these cross section(s) are decreased slightly until reaching the minimum safe cross section(s), then the fitness will be improved.

30

Chapter (3)


In addition, the chromosome length needed for representing sizing variable depends on the number of discrete available cross sections equation (3.3). So if the available cross sections are few, the solution will not be accurate enough and if the available sections are many the chromosome length will be long which means more proposed solution and more time consumed. In addition, any little change in truss topology means adding or removing member(s) which leads to huge change in the proposed truss. In contrast, size and shape variables have lower and upper limits and graduated values inbetween. So it is complicated to make size, shape and topology optimization at the same time. Also making this full optimization leads to a large number of possible solutions which needs long time consumed. To overcome these drawbacks, a proposal is suggested to make topology and shape optimization only and get cross sections area from design. We have to start with suggesting that A/L = Constant value for all truss members to be able to substitute in {F}=[K]*{D} equation. After that we have to increase/decrease A/L to get the minimum safe cross sections which satisfy constrains. In that case these cross sections are not the suggested cross sections but they are final safe cross sections. However, Using cross section as a variable (traditional method) produces better result considering 'A/L' varied. This gives the opportunity to use the optimal cross sections. So new method is needed to avoid traditional drawbacks besides keeping 'A/L' varied. 31

Chapter (3)


3.4 The proposed approach As shown the traditional method drawbacks called for the need to develop new method using new developed variables to avoid these drawbacks while keeping stiffness A/L varied between different members. The proposed idea is to use design variables only for nodes deflections and coordinates along with strain condition.

3.4.1 Proposed Design Variables The proposed approach is developed to overcome the drawbacks of using member cross sections as a variable (random case and long chromosomes). Deflection is used as a variable instead of member cross sections. While topology optimization is made by keeping members have elongation more than zero and less than allowable strain and remove the others. In other words proposed approach is using nodes deflections for non-support nodes and nodes coordinates for coordinately variable nodes as variable along with strain condition to make topology, sizing and shape optimization at the same time. This means that the design variables are now reduced to two or three coordinates for each coordinately variable node and two or three deflection components for each non-support node. Referring to the same example of the 10 -bar plane truss figure (3.1) the chromosome will consist of two parts:

32

Chapter (3)


A- For Shape Number of bites for shape optimization is Similar to traditional case. Number of bites of X-coordinates of one node = 8 bites. Number of bites of Y-coordinates of one node = 7 bites. B- For Deflection Let the desired precision be 1 mm. Deflection variable limits range from 0.0 mm to -50.8 mm (input data of 10-bar plane truss). Number of bites for deflection variable of one node in X or Ydirection = log 2 [

0−(−50.8) 1

] = 5.66 = 6 bites.

Combining Chromosome Total number of Chromosome bites = (number of bites for Xcoordinates + number of bites for Y-coordinates) × number of variable nodes + (number of bites for deflection in X&Y direction)× Total number of non-support nodes. Total number of Chromosome bites= (7+8)× 2+ (6+6)× 4 =78 bites.

33

Chapter (3)


Table (3.1) illustrates number of bites used in traditional and proposed methods where proposed approach uses 78 bites while Traditional one uses 120. Table 3.1: Comparison between number of bites in traditional and proposed approach Optimization Variable

Traditional Method

Shape

30

30

Size

80

‫ـــــــــــــــــــــ‬

Topology

10

‫ـــــــــــــــــــــ‬

Deflection

‫ــــــــــــــــــــــ‬

48

Total

120 Bites

78 Bites

1.32923E+36

3.02231E+23

No. of possible solutions

Proposed Method

Note: proposed approach used shape (coordinates) and deflection variables for making shape, size and topology optimization at the same time.

Also, the number of possible bites combinations, i.e. possible solutions can be calculated using equation (3.5) where two options are available for each bite. it is shown that number of possible solutions has been reduced from 1.32923E+36 to 3.02231E+23. 𝑁𝑜. 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 = 2𝑛

(3.5)

where n is number of bites in the individual chromosomes combination.

3.4.2 Extension to Sizing and Topology Design variables of the truss in this approach are coordinates of each node (xi, yi for plane truss and xi, yi, zi for space truss, where i=1 to N, N is the number of truss coordinately variable nodes). The range of nodal coordinates are predefined or defined by the truss 34

Chapter (3)


proposed geometric limits considering constructability constrain besides deflections of each node (ui, vi for plane truss and ui, vi, wi for space truss, where i=1 to N, N is the number of truss non-support nodes). The range of nodal deflections is defined by the code limits of allowable deflection. Figure (3.2) shows chromosome sample for the ten bars plane truss.

Figure 3.2: Chromosome sample for 10-bar plane truss.

After generating chromosome for these variables and translating these chromosomes, the proposed node coordinates for free nodes and node deflections for all non-support nodes to combine the proposed truss are obtained. Primary Topology Matrix is predefined or considered from input data which describes all logical possible distribution of members between different nodes in the truss shown in figure (3.1).

35

Chapter (3)


Each row represents a single member. The first column represents member number from 1 to m where m is the number of truss members. Columns No. 2, 3 and 4 represent the degrees of freedom of the start node of the member and columns No. 5, 6 and 7 represent the degrees of freedom of the end node of the same member. Using

design

variables

group

which

represent

nodes

coordinates, the lengths of each member (Li) can be calculated as following: Li = √∆x 2 + ∆y 2 + ∆z 2

(3.6)

Using the other group of design variables which represent nodal deflections, the deformed length of each truss member can be calculated. The change in length (L) can be derived using the relation:

36

Chapter (3)


ΔL = ∆x Cosθx + ∆y Cosθy + ∆z Cosθz

(3.7)

where x, y, z are the direction angles of the truss member defined by Cosθx =

x2 − x1 Li

(3.8)

Cosθy =

y2 − y1 Li

(3.9)

Cosθz =

z2 − z1 Li

(3.10)

and x, y, z are the net displacements of the member defined as: ∆x = u2 − u1

(3.11)

∆y = v2 − v1

(3.12)

∆z = w2 − w1

(3.13)

where xi, yi, zi, ui, vi, wi are the nodal coordinates and displacements of member joints, respectively. The strain of each member is then compared to the allowable strain of used material (all) as: 0.0
1e−9

(3.16)

where |K| is the determinant of stiffness matrix for the proposed truss. If the proposed truss satisfies equation (3.16), the structure is stable and the program proceeds in calculating the fitness. If equation (3.16) is not satisfied, the truss will be unstable and the proposed solution will be refused.

3.4.4.2 Constraint 2: Constructability This constrains is taken into consideration by preventing two members from having the same start and end nodes and by not allowing two or more nodes to have same coordinates. It can be achieved in 10-bar plane truss figure (3.1) by considering the proposed X-coordinates of node No. 6 larger than the proposed Xcoordinate of node No. 5. (not equal or less).

42

Chapter (3)


3.4.4.3 Constraint 3: Member stresses Member stresses restrictions are the most important criteria in structural engineering. Member stresses must not exceed a certain limit as following. σi ≤ σall

(3.17)

where σi is the stress for member i; and σall is the allowable stress. Otherwise the truss members will fail and this means that the truss is unable to support the loads and the structure is unsafe. Stress constrains is considered in study cases of benchmark problems according to input data of each problem If the proposed truss satisfies equation (3.17), the member stresses don't violate stress constrain and the proposed truss will be accepted. Otherwise, the member stresses violate stress constrain and the proposed truss will be refused.

3.4.4.4 Constraint 4: Nodal displacements Displacement restrictions are often crucial in structural engineering. The structure is not allowed to deflect more than a certain limit as following. δi ≤ δall

(3.18)

where δi is the deflection for node i; and δall is the allowable deflection.

43

Chapter (3)


If the proposed truss satisfies equation (3.18), the nodes deflections don't violate deflection constrain and the proposed truss will be accepted. Otherwise, the nodes deflection violate deflection constrain and the proposed truss will be refused. Stability and construability are necessary in case of making shape or/and topology optimizations. However, they are not needed in case of making only sizing optimization because in that case node coordinates and member connectivity are constant. On the contrary, stress and deflections constrains are necessary for any optimization category.

3.5

The Proposed algorithm The proposed optimization algorithm is a bit-string encoded

GA. Mutation and two point crossover type is used as reproduction operators. The master flow chart of the proposed approach is shown in figure (3.3). It starts with input data such as primary topology matrix, fixed nodes (load nodes and support nodes) coordinates, global load vector, allowable stress, allowable deflection, number of populations,

number

of

generations

and

mutation/crossover

probabilities. The next step is creating initial population according to number of population which is given in input data. Initial population is proposed values for coordinates of non-fixed nodes and deflections for non-support nodes. Each proposed solution is represented in 44

Chapter (3)


binary coded chromosome. At this step, constructability constrain is considered by avoiding any two nodes have the same coordinates. The following process is translating this binary coded chromosome to decimal values. The proposed truss can be assembled by using these decimal values for coordinates and deflection variables. By applying strain condition, primary topology matrix will be updated to produce optimized one. So at this step, the proposed optimized truss (shape and topology) is combined. The subsequent process is making full analysis for the whole proposed optimized truss considering 'A/L=constant' to get the internal forces in each member. The next step is checking the stability for this proposal optimized truss. The proposed truss must pass stability constrain otherwise, it will be considered unfit proposed solution. The following operation is calculating proposed cross sections for truss members using design criteria equation (3.15) where these proposed cross sections is almost optimal cross sections. At this stage, the whole proposed optimized truss (shape, size and topology) is formed where this proposed truss is optimal or near optimal truss according to proposed coordinates. The next procedure is final and full analysis for this proposed optimized truss to get the member stresses and nodes deflections. At this stage, stress and deflection constrains are checked by comparing the resulted member stresses and node deflections with allowable 45

Chapter (3)


values of stress and deflections. The truss must pass stress and deflection constrains to be valid truss otherwise, it will be unfit proposed solution. For any solution violates any constrains will take penalty to exclude this solution. On the contrary, fitness function is calculated for solution passed all constrains. Finally, if the number of generations reaches given limit the calculation processes will stop to start drawing output best truss. Otherwise, new generation will deformed using cross over and mutations probabilities.

46

Chapter (3)

The Proposed Approach Input data

Create initial population Input data For Each chromosome

Create initial population Assemble the basic structure and translate the For Each chromosome binary chromosome in to FEM structure

Update Assemble the basic topology structure matrix and translate the binary chromosome in to FEM structure Analyze the truss suggesting A/L=Const. Update topology matrix Check on the stability Analyze the truss and get cross sections No

Passed and stability Check on constructability Yes No

Calculate the suggested sections

Passed Yes

Calculate Elementthe Stresses and Node Deflection Calculate suggested sections Penalty any violations Calculate Element Stresses and Node Deflection

Evaluate Fitness Penalty anythe violations Reached

Termination Evaluate the Fitness Criteria Not Reached

Reached

Termination Cross over & Mutation Criteria Not Cross over & Mutation Selection & New Generation Reached

Translate the binary chromosome and draw the truss and results

Figure

Translate the binary chromosome andThe drawmaster the truss and results 3.3: flow chart

approach 47

of the proposed.

Chapter (4)

Case Study And Results

Chapter (4) CASE STUDY AND RESULTS 4.1 General Structure optimization is very important in structural design process especially in truss design. It aims to produce the most optimum safe truss with minimum weight. In this chapter, the proposed approach is applied to benchmark problems to evaluate its capabilities through comparison with studies conducted in the literature. The first problem is size shape and topology optimization for benchmark ten bar plane truss which is commonly used to verify new truss optimization approach. The second optimized size benchmark 25-bar space truss and extended to size shape and topology optimization for the same truss. The third problem is size optimization for benchmark 72-bar space truss.

4.2 Benchmark Ten bar Plane Truss The first problem is shape, sizing and topology optimization for ten bar plane truss. This ten-bar truss is often used as a benchmark problem in structural optimization. This structure is frequently found in literature related to plane truss optimization [3, 5]. The truss has two vertical supports with distance 'a' of 9.144 meters (360 inches) and two loads 'F'of 445.374 KN (100 kips) at 9.144 and 18.288 meters from the lower support as shown in figure (4.1). Aluminum is used, with Modulus of elasticity E equals 68.95 GPa (104 ksi), density ρ equals 2, 768 kg/m3 (0.1 lb/in3) and element stresses are 48

Chapter (4)


limited to 172.37 MPa (25 ksi) in both tension and compression while buckling and own weight are ignored. The displacements are limited to 50.8 mm (2 in) both horizontally and vertically. Shape is optimized for coordinates of nodes P1 and P3 only within geometry limits of the truss while support and load nodes are fixed.

Figure 4.1: Structure of benchmark ten bar plane truss.

These input data and constrains are similar to those used in previous works for ten bar plane truss to make fair comparison between this study and previous works. The problem was solved using GA by Rajeev [20], Coello [15], Galante [37], Ruiyi, GuiLiangjin, Zijie [38], Schmid [39], Rajan [23], Hajela and Lee [22], Xu [40], Rajeev [24], Wenyan [41], Rahami [42] and Gulati [26]. It was also solved by Baugh Jr. et al. [43] using hybrid search method. Moreover, it was solved by Liu et al. [6] using particle swarm. 49

Chapter (4)


Using the suggested technique, weight is minimized utilizing GA with parameters as following: Population size is 600 which it is sufficient to show the capability of the proposed approach for this example, maximum generation 200 which it is enough to obtain best fitness for this population size, crossover probability equals 0.9 and mutation probability equals 0.05. Available cross sections are similar to cross sections (226 cross sections) were used by Hultman [3]. As mentioned earlier, number of available cross section doesn't affect chromosome length or GA efficiency. As illustrated in chapter 3, 226 cross sections were used for making full optimization for 10-bar plane truss. It consumed 120 bites using traditional while 78 bites only were consumed using current approach Figure (4.2) shows the convergence history of the ten bars plane truss example. The mean fitness value is plotted against the generation number to clarify how the GA converges to the optimum solution. Two population sizes were used. These are 400 and 600 where 400 produced truss weighs 4824 lb. while 600 produced truss weighs 4762.1 lb. Using population size more than 600, there is not reduction in truss weight. As depicted from the figure, the fitness value (which reflects the truss weight taking into consideration violated constrains ) improvement is very limited during the first 30 generation, and then greatly improved till the 100th generation after which stability is observed. 50

Chapter (4)


Figure 4.2: Convergence history of benchmark ten bar plane truss structure.

Figure (4.3) shows the shape of optimized truss. This optimized topology is similar to that resulted in all researches which made topology along with shape optimization.

51

Chapter (4)


Figure 4.3: Optimized structure of the benchmark ten bar truss for current. study.

Nodal coordinates for coordinately variable node P3 is (11.73, 6.4) while node P1 is deleted. Table (4.1) shows member cross section areas and their stresses. Maximum stress absolute value for our resulted optimized truss is 161.50 MPa in member M1 which represents 93.7% of allowable stress. And the maximum deflection absolute value investigated in the optimum solution is 50.763 mm for node P2 in Y direction which reaches 99.92% of the maximum permissible value which means that the selected cross sections are almost optimized.

52

Chapter (4)


Table 4.1: Optimization results for 10-bar plan truss.

Box section Element

dimensions, (mm)

Cross section

Stress,

% of allowable

area, (cm2)

(MPa)

stress

M1

200x200x5

39.00

161.5

93.70%

M2

400x400x10

156.00

-57.9

33.60%

M3

260x260x12.5

123.74

-39.9

23.10%

M4

180x180x12

80.65

-56.8

32.90%

M5

350x350x10

136.00

46.97

27.20%

M6

400x400x12

186.26

49.12

28.50%

Verification of the proposed procedure was carried out using SAP2000 program to double check for the deflection. This was conducted because this example is controlled by deflection; i.e. deflection has reached allowable value (99.92%) before stress reached its allowable value. The applied loads of the final optimized truss are shown in figure (4.4) while the maximum nodal displacement is shown in figure (4.5). The maximum actual deflection absolute value as shown is 50.796 mm while our MATLAB code produced 50.763 mm for node P2 in Y direction which is approximately matched.

53

Chapter (4)


Figure 4.4: Load case of the ten bar truss in SAP Program (N).

Figure 4.5: Maximum deflection of the ten bar truss in SAP Program (mm). (N,mm). 54

Chapter (4)


Table (4.2) lists the results encountered in literature for the 10 member benchmark problem compared with the results of the proposed approach of the present study. The results shows that the current proposed approach resulted in the most optimized value of fitness (weight) which is less than all results found in the literature except those obtained by Deb and Gulati [26] where they obtained a better solution (4731.65 lb.) when the constructability condition wasn't considered. Figure (4.6) shows their optimum solution which contains two duplicated members. One member was between nodes P6 & P4 and the other member was between nodes P6 & P2. This is considered undesirable overlap and unpractical to implement [26].

Figure 4.6: Optimized structure of the 10-bar truss Deb and Gulati [15] (4731.65 Lb).

When they considered constructability condition, i.e. preventing members overlapping, they obtained a truss weight of (4899.15 lb.) and produced same topology produced in the current study. Most of literature studies considered limited optimization categories i.e. shape and sizing or topology and sizing or only size 55

Chapter (4)


optimization. That is to avoid the complexity resulted from making full optimization (size, shape, and topology at the same time). Rajeev [24] made size, shape and topology optimization at the same time, but used limited topology optimization and conducted the analysis using two stages as mentioned earlier in Chapter 2. This is different from the current approach which considers size, shape and topology optimization at the same time, through one stage and without any limitation on topology optimization. Schmid [39] made size and shape optimization for topology optimized truss (6-bar truss) as shown in figure (4.3).

56

Chapter (4)


Table 4.2: Results of previous works with same conditions. Optimization Search

Weight

category

Method

(lbs.) Size

Shape

Topology

Rajeev (1992)

LINRM

√

6249.00

Rajeev (1992)

SUMT

√

5932.00

Rajeev (1992)

GRP-UI

√

5727.00

Rajeev (1992)

M-5

√

5725.00

Rajeev (1992)

M-3

√

5719.00

Genetic algorithm

√

5613.84

Coello (1994)

Genetic algorithm

√

5586.59

Rajeev (1992)

CONMIN

√

5563.00

Rajeev (1992)

OPTDYN

√

5472.00

Galante (1996)

Genetic algorithm

√

Xu (2010)

Genetic algorithm

√

5078.23

Kripakaran et al. (2007)

Hybrid method.

√

5073.03

Li et al. (2006)

Particle swarm

√

5060.9

Su Ruiyi et al. (2009)

Genetic algorithm

√

Schmid (1997)

Genetic algorithm

√

Rajan (1995)

Genetic algorithm

√

√

4962.1

Hajela (1995)

Genetic algorithm

√

√

4942.7

Rajeev (1997)

Genetic algorithm

√

√

4925.80

Wenyan (2005)

Genetic algorithm

√

√

4921.25

Rahami (2008)

Genetic algorithm

√

√

4855.2

Rajeev (1992)

√

5119.3

√ √

√

Note

4962.07 4962.10

Two stages

Const. Genetic algorithm

√

√

4731.65

constrain not considered

Gulati (2001) Const. Genetic algorithm

√

√

4899.15

constrain considered

Current study (2015)

Genetic algorithm

√

57

√

√

4762.1

Chapter (4)


4.3 Benchmark 25-Bar Space Truss The second benchmark problem is 25-bar space truss optimization. This structure is frequently found in the literature related to space truss optimization [44]. Most of the works found in the literature are related to sizing optimization for which node location and connectivity are already known. In current work, at first sizing optimization was applied to compare with the results of previous works. Then the full optimization procedure was applied to obtain the most optimum shape, topology and sections. This simultaneous optimization for 25-bar space truss has also been conducted by Wenyan [41] and Rahami [42] where shape optimization was made for nodes number 3, 4, 5, 6, 7, 8, 9 and 10 as shown in figure (4.7).

4.3.1. Sizing optimization of the 25-bar space truss Figure (4.7) shows the truss shape. The coordinates of joints, the member groups for section selection and applied loads are shown in tables (4.3), (4.4) and (4.5) respectively. Aluminum is used with modulus of elasticity E equals 68.95 GPa (104 ksi) and density, ρ equals 2,768 kg/m3 (0.1lb. /in3) and element stresses are limited to 275.8 MPa (40 ksi) in both tension and compression while buckling and own weight are ignored. The displacements of nodes are limited to 8.9 mm (0.35 in) in all directions.

58

Chapter (4)


Figure 4.7: Benchmark 25-bar space truss structure (inch=2.54 cm).

59

Chapter (4)


Table 4.3: Coordinates of the joints of the 25-bar space truss.

Node 1 2 3 4 5 6 7 8 9 10

X (m) -0.9525 0.9525 -0.9525 0.9525 0.9525 -0.9525 -2.54 2.54 2.54 -2.54

Y (m) 0 (Inch) 0 0.9525 0.9525 -0.9525 -0.9525 2.54 2.54 -2.54 -2.54

Z (m) 5.08 (Inch) 5.08 2.54 2.54 2.54 2.54 0 0 0 0

Table 4.4: Group membership for 25-bar space truss.

Group Number 1 2 3 4 5 6 7 8

Members 1-2 1-4, 2-3, 1-5, 2-6 2-5, 2-4, 1-3, 1-6 3-6, 4-5 3-4, 5-6 3-10, 6-7, 4-9, 5-8 3-8, 4-7, 6-9, 5-10 3-7, 4-8, 5-9, 6-10

Table 4.5: Loading conditions for 25-bar space truss. Node

Fx (KN)

Fy (KN)

Fz (KN)

1

4.45374

-44.5374

-44.5374

2

0

-44.5374

-44.5374

3

2.22687

0

0

6

2.672244

0

0

These similar to data input and constrains are used in previous works.

60

Chapter (4)


The problem was solved using GA by Rajeev [20], Zhu [45], Erbatur et al. [46], Coello et al. [47], Cao [48], Togan [49], Tayfun [44], Wu [50], Wenyan [41] and Rahami [42]. It was solved by Talaslioglu [51] using Bi population-Based GA with enhanced interval search. It was also solved by Li et al. [9] using heuristic particle swarm optimization. It was solved by Lee et al. [52] using Harmony Search Heuristic Algorithm. It was also solved by Camp [8] using Bang –Crunch Algorithm. It was also solved by Kaveh and Shojaee [7] using Ant Colony Optimization. All these pre-mentioned researches made sizing optimization only for 25-bar space truss. The set of available cross section areas for this problem is A=i*0.64516 (cm2) where i ranges from 1 to 50. Using this technique weight is minimized by GA with parameters as following: Population size is 400 where it is the minimum adequate value to show the capability of the proposed approach for this example, maximum generation 100 where it is enough to reach best fitness for this population size, crossover probability equals 0.9 and Mutation probability equals 0.05. Figure (4.8) shows the model of 25-bar space truss in MATLAB while Figure (4.9) shows the convergence history of the 25-bar space truss example. The fitness mean value is plotted against the generation number to clarify how the GA converges to the optimum solution. As obvious from the plot, the fitness value is greatly improved from first generation till the 60th generation after which stability is observed. 61

Chapter (4)


Figure 4.8: 25-bar space truss structure model in MATLAB.

62

Chapter (4)


Figure 4.9: Convergence history of 25-bar space truss. structure.

Table (4.6) shows the resulted member cross section areas and their stresses where maximum absolute stress value is 124.17 MPa (18.01 ksi) in member 6-3 which represents 45.02% of allowable stress. Moreover, maximum absolute deflection value in the optimum solution is 8.89 mm for node 1 in Y direction which reaches 99.89% of the maximum permissible value meaning that the selected cross sections are almost economic.

63

Chapter (4)


Table 4.6: Member stresses for 25-bar space truss.

Member

Area (cm2)

Length (m)

Weight (kg)

Axial Force (KN)

Abs. stress value (MPa)

1-2

0.65

1.91

0.34

-0.03

0.48

2-6

1.94

3.31

1.78

16.97

87.68

1-5

1.94

3.31

1.78

13.88

71.71

2-3

1.94

3.31

1.78

3.15

16.29

1-4

1.94

3.31

1.78

4.51

23.30

1-6

23.23

2.71

17.44

248.06

106.80

1-3

23.23

2.71

17.44

45.53

19.60

2-5

23.23

2.71

17.44

204.28

87.95

2-4

23.23

2.71

17.44

66.87

28.79

6-5

10.97

1.91

5.78

-29.94

27.30

3-4

10.97

1.91

5.78

13.85

12.63

5-4

0.65

1.91

0.34

-5.44

84.25

6-3

0.65

1.91

0.34

-8.01

124.17

4-9

5.16

4.60

6.57

3.31

6.40

5-8

5.16

4.60

6.57

-14.70

28.48

6-7

5.16

4.60

6.57

-20.53

39.78

3-10

5.16

4.60

6.57

6.01

11.63

6-9

2.58

4.60

3.29

-10.27

39.78

5-10

2.58

4.60

3.29

-7.35

28.48

4-7

2.58

4.60

3.29

1.65

6.41

3-8

2.58

4.60

3.29

3.00

11.63

5-9

23.23

3.39

21.79

-89.77

38.65

6-10

23.23

3.39

21.79

-125.40

53.99

4-8

23.23

3.39

21.79

20.19

8.69

3-7

23.23

3.39

21.79

36.67

15.79

216.061

Max. stress

124.174

Weight (kg)

64

Chapter (4)


Verification of the proposed procedure was carried out using SAP2000 program to check the deflection since this example is controlled by deflection. Figure (4.10) shows the load case and figure (4.11) shows the maximum absolute deflection value which is 8.8002 mm while MATLAB produced 8.89 mm for node 1 in Y direction which is approximately matched.

Figure 4.10: Load case of the 25-bar space truss in SAP program (KN).

65

Chapter (4)


Figure 4.11: Maximum deflection of the 25-bar space truss in SAP program (mm).

Table (4.7) lists the results encountered in the literature for only sizing optimized 25 member space truss problem compared with the results of the proposed approach. As shown in the results, the proposed approach resulted in the most optimized value of fitness (weight) which is less than all results found in literature. the sizing optimized truss weight is 476.33 lb. where it represents 98.54% of the best result encountered in the literature (Togan [49]).

66

67

1.8

2.3

0.2

0.1

0.8

1.8

3

546.01

A2

A3

A4

A5

A6

A7

A8

Weight (lbs.)

562.93

2.6

2.1

0.8

0.1

0.1

2.6

1.9

0.1

1986

493.8

3.4

0.4

0.9

0.1

0.1

3.2

1.2

0.1

2000

[46]

493.94

3.4

0.5

1.1

1.4

0.1

3.2

0.7

0.1

1994

[47]

485.05

3.4

0.5

0.9

1.9

0.1

3.4

0.5

0.1

1996

483.35

3.4

0.5

1

2

0.1

3.4

0.3

0.1

2008

[49]

485.9

3.4

0.7

1

1.9

0.1

3.4

0.1

0.1

2009

BGAwEIS

[51]

484.85

3.4

0.5

1

2.1

0.1

3.4

0.3

0.1

2009

HPSO

[9]

Talaslioglu Li et al.

484.85

3.4

0.5

1

2.1

0.1

3.4

0.3

0.1

HSH 2005

[52]

al.

Lee et

484.85

3.4

0.5

1

2.1

0.1

3.4

0.3

0.1

2007

BB-BC

[8]

Camp

484.85

3.4

0.5

1

2.1

0.1

3.4

0.3

0.1

2007

[7] ACO

Kaveh

484.85

3.4

0.5

1

2.1

0.1

3.4

0.3

0.1

2011

[44]

Tayfun

Interval Search.

Optimization, HSH: Harmony Search Heuristic Algorithm, BGAwEIS: Bi population-Based Genetic Algorithm with Enhanced

BB-BC: Big Bang –Crunch Algorithm, Algorithm, ACO: Ant Colony Optimization, HPSO: Heuristic Particle Swarm

0.1

1992

[20]

A1

(area in2)

Design variable Rajeev

Zhu [45] Erbatur Coello Cao [48] Togan

Table 4.7: Sizing optimized 25-bar space truss results.

476.33

3.6

0.4

0.8

1.7

0.1

3.6

0.3

0.1

2015

study

Current

Chapter (4) Case Study And Results

Chapter (4)


4.3.2. Sizing, shape and topology optimization of the 25-bar space truss. To perform full optimization of the same problem, using the size, shape and topology where optimized simultaneously using the same data input, constrains, and GA parameters. For shape optimization case, the coordinates limits for coordinately variable node number 4 in X-direction From 20 to 60 inch and in Y-direction from 40 to 80 inch and in Z-direction from 90 to 130 inch where another variable nodes 3,5 and 6 are considered symmetric to node number 4. Moreover, for support node number 8, in X-direction from 40 to 80 inch; and in Y-direction from 100 to 140 in other support nodes number 7, 9 and 10 are considered symmetric to node number 8. Figure (4.12) shows the resulted topology, sizing and shape optimized 25-bar space truss in MATLAB. It is worthwhile to note that 6 members are removed (Groups number 1, 4, 5 and one member from group number 2). Figure (4.13) shows the convergence history of the 25 bars space truss example, where the mean fitness value is plotted against the generation number. It is clear that the fitness value greatly improves from the first generation up to the 40th generation after which stability is observed. By comparing with convergence history of only sizing optimization case, it is shown that the optimum solution is reached more early in case of simultaneous sizing, shape and topology optimization. This is due to the fact that using node coordinates along with its 68

Chapter (4)


deflections allows changing in member length and its elongation and leads to more options in topology optimization which allows removing unnecessary members.

Figure 4.12: Shape, sizing and topology optimized 25-bar space truss structure model in MATLAB.

69

Chapter (4)


Figure 4.13: Convergence history of 25-bar Space truss structure.

Table (4.8) shows the coordinates of coordinately variable nodes number 3, 4, 5 and 6 along with coordinates of support nodes number 7, 8, 9 and 10. It is shown that the truss turned into wideness in Y-direction.

70

Chapter (4)


Table 4.8: Coordinates of coordinately optimized nodes of 25-bar space truss.

Node

X (m)

Y (m)

Z (m)

3

-1.0312

1.4834

3.1445

4

1.0312

1.4834

3.1445

5

1.0312

-1.4834

3.1445

6

-1.0312

-1.4834

3.1445

7

-1.4275

3.5357

0.00

8

1.4275

3.5357

0.00

9

1.4275

-3.5357

0.00

10

-1.4275

-3.5357

0.00

Table (4.9) shows the resulted member sections and their stresses where maximum absolute stress value is 119.62 MPa (17.35 ksi) in member 2-5 which represents 43.73% of allowable stress and maximum absolute deflection value is 8.8975 mm for node 1 in the Y direction which reaches 99.97% of the maximum permissible value which means that the selected cross sections are almost economic.

71

Chapter (4)


Table 4.9: Member stresses of optimized 25-bar space truss. . Membe r 1-2 2-6

Area (cm2)

Length (m)

Weight Axial Force (kg) (KN) Removed 0.56 3.76

Abs. stress value (MPa)

0.65

3.14

1-5

0.65

3.14

0.56

6.89

106.82

2-3 1-4 1-6

0.65

3.14

0.56

3.33

51.61

5.81

2.44

3.92

53.99

92.98

1-3

5.81

2.44

3.92

33.98

58.52

2-5

5.81

2.44

3.92

69.47

119.62

2-4 6-5 3-4 5-4 6-3 4-9

5.81

2.44

3.92

28.18

48.53

0.65

5.94

1.06

1.56

24.22

5-8

0.65

5.94

1.06

0.32

5.01

6-7

0.65

5.94

1.06

0.86

13.34

3-10

0.65

5.94

1.06

1.77

27.50

6-9

0.65

4.49

0.80

1.14

17.64

5-10

0.65

4.49

0.80

0.43

6.63

4-7

0.65

4.49

0.80

2.07

32.04

3-8

0.65

4.49

0.80

2.35

36.38

5-9

6.45

3.78

6.74

5.08

7.87

6-10

6.45

3.78

6.74

13.53

20.97

4-8

6.45

3.78

6.74

24.57

38.09

3-7

6.45

3.78

6.74

27.90

43.24

51.79

Max Stress

119.62

58.21

Removed

Removed Removed Removed Removed

Weight (kg)

Verification of the proposed procedure was carried out using SAP2000 program to check the deflection since this example is controlled by deflection. Figure (4.14) shows the load case and figure (4.15) shows the maximum absolute deflection absolute value which is 8.8889 mm while MATLAB produced 8.8975 mm for node 1 in Y direction which is approximately matched. 72

Chapter (4)


Figure 4.14: Load case of the 25-bar space truss in SAP program (KN).

Figure 4.15: Maximum deflection of the 25-bar space truss in SAP program (mm).

73

Chapter (4)


Table (4.10) shows the literature results compared with the proposed approach result. As seen from the table, the current proposed approach resulted in the most optimized value of fitness (weight) which is less than all results found in literature. For sizing, shape and topology optimized truss the weight is 114.171 lb. where it represents 99.83% of the best result encountered in the literature (Rahami [42]). As shown in table (4.10), Wenyan [41], Rahami [42] and current study removed member groups number 1, 4 and 5. But the current study removed one extra member from group number 2 while the other members in the same group are remained. It proves the capability of making topology optimization using strain condition along with deflections and coordinates variable. Also it shows the sensitivity of this method for removing all members which can be removed. The weight of sizing optimized 25-bar space truss using current method is 476.33 lb, while it is 114.171 lb using same method for optimizing size, shape and topology of the same truss. The reduction in weight (76.03% reduction) shows the importance of making full (size, shape and topology) optimization for trusses.

74

Chapter (4)


Table 4.10: Size, shape and topology optimized 25-bar space truss results.

Wu [50] Members and its groups

1-2 2-6 1-5 2-3 1-4 1-6 1-3 2-5 2-4 6-5 3-4 5-4 6-3 4-9 5-8 6-7 3-10 6-9 5-10 4-7 3-8 5-9 6-10 4-8 3-7

Group 1

Group 2

Group 3

Group 4 Group 5

Group 6

Group 7

Group 8

X4 (in) Y4 (in) Z4 (in) X8 (in) Y8 (in) Weight (lb.)

1995 0.1 0.2 0.2 0.2 0.2 1.1 1.1 1.1 1.1 0.2 0.2 0.3 0.3 0.10 0.10 0.10 0.10 0.2 0.2 0.2 0.2 0.90 0.90 0.90 0.90 41.07 53.47 124.6 50.8 131.48 136.2

Wenyan Rahami Current study [41] [42] 2005 2008 2015 2 Cross section area (in ) Removed Removed Removed 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 Removed 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 Removed Removed Removed Removed Removed Removed Removed Removed Removed Removed Removed Removed 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 39.91 38.7913 40.60 61.99 66.111 58.40 118.23 112.9787 123.80 53.13 48.7924 56.20 138.49 138.891 139.20 114.74 114.3701 114.171

75

Chapter (4)


4.4 Benchmark 72-Bar Space Truss The third problem is size optimization for 72-bar space truss. This truss consists of 4 support nodes and four symmetric levels. Each level has 4 free nodes and 18 members as shown in figure (4.16). This structure is frequently found in literature related to space truss optimization [44]. Allowable stress equals 172.37 MPa (25 ksi) for all members from 1 to 72 in both tension and compression while buckling and own weight are ignored. Allowable displacement equals 6.35 mm (0.25 in) in only X and Y directions for only nodes numbers 17, 18, 19 and 20. Modulus of elasticity E equals 68.95 GPa (104 ksi) and Density of material equals 2, 768 kg/m3 (0.1 lb. /in3). Members grouping are shown in table (4.11). These input data and constrains are similar to those used in previous works for 72-bar space truss to make fair comparison between this study and previous works. Two load cases are considered as shown in table (4.12). In the first load case, the truss is controlled by deflection constrain where in the second one, the truss is controlled by stress constrain.

76

Chapter (4)


Figure 4.16: Benchmark 72-bar truss structure (a) profile view, (b) element and node numbering system.

77

Chapter (4)


Table 4.11: Members grouping for 72-bar space truss.

Group No.

Group Members

Group No.

Group Members

A1

1,2,3,4

A9

37,38,39,40

A2

5,6,7,8,9,10,11,12

A10

41,42,43,44,45,46,47,48

A3

13,14,15,16

A11

49,50,51,52

A4

17,18

A12

53,54

A5

19,20,21,22

A13

55,56,57,58

A6

23,24,25,26,27,28,29,30

A14

59,60,61,62,63,64,65,66

A7

31,32,33,34

A15

67,68,69,70

A8

35,36

A16

71,72

Table 4.12: Load cases for 72-bar space truss. Load Case

Node

Fx (KN)

Fy (KN)

Fz (KN)

1

17

22.24

22.24

-22.24

17

0.00

0.00

-22.24

18

0.00

0.00

-22.24

19

0.00

0.00

-22.24

20

0.00

0.00

-22.24

2

The problem was solved using GA by Adeli and Kamal [53], Cao [48], Venkayya [54], Schmit and Farshi [55], Khan et al. [56], Erbatur et al. [46], Gellatly and Berke [57], Barbaso and Lemonge [58] Ayvaz and Aydın [59], Dede et al. [60] and Tayfun [44]. It was also solved by Talaslioglu [51] using BipopulationBased GA with enhanced interval search. It was solved also by Camp [8] using Big Bang-Crunch Algorithm; and was solved by

78

Chapter (4)


Perez and Behdinan [61] using Particle Swarm Optimization Algorithm. Cross section variable is continuous as it is used in most of literature studies. Continues variable in traditional method has upper and lower limits besides precision for the graduated values inbetween these limits as it shown in equation (3.3). These limits and precision values affect chromosome length which represents the variable. But in current study, continuous cross section means cross sections which are obtained directly from design equation (3.15). So the current approach doesn't need limits or precision while the equation produces cross sections have infinite accuracy without any increment in chromosome length. Using this technique weight is minimized by GA with parameters as following: Population size is 800 where it is the minimum adequate value to show the capability of the proposed approach for this example; maximum generation 200 where it is enough to reach best fitness for this population size, crossover probability equals 0.9 and mutation probability equals 0.05. Two deflection variables are used for each node one is for load case 1 and the other is for load case 2. Table (4.13) shows the stresses in each member of 72-bar truss for both load cases. Figure (4.17) shows the convergence history of the 72 bars space truss

79

Chapter (4)


example. The mean fitness value is plotted against the generation number to clarify how the GA converges to the optimum solution. As investigated from the plot, the fitness value is greatly improved from first generation till the 150th generation after which stability is observed. Figure (4.18) shows the 72-bar space truss model in MATLAB program.

Figure 4.17: Convergence history of 72-bar space truss structure.

80

Chapter (4)


Figure 4.18: Benchmark 72-bar truss structure Model in MATLAB Program.

81

Chapter (4)


For load case 1 Maximum absolute stress value is 111.6 MPa (16.186 ksi) for member 55 which represents 64.74% of allowable stress. Maximum absolute deflection value investigated in the optimum solution is 6.34492 mm for node 17 in X and Y direction which represents 99.95 % of allowable deflection. Verification of the proposed procedure was carried out for load case 1 using ETABS program to check deflection since this load case is controlled by deflection. Figure (4.19) shows the load case and figure (4.20) shows the maximum absolute deflection value which is 6.3 mm while MATLAB produced 6.34492 mm for node 17 in X and Y direction which is approximately matched.

82

Chapter (4)


Figure 4.19: Load case 1 for 72-bar truss structure in ETABS program (N).

83

Chapter (4)


Figure 4.20: Maximum deflection of load case 1 for 72-bar truss structure in ETABS program (mm).

For load case 2 Members 55, 56, 57 and 58 got maximum stress value 167.40 MPa (24.28 ksi) which represents 97.144% of allowable stress. While nodes 17, 18, 19 and 20 got maximum deflection value 0.99949 mm in X and Y directions which represents 15.74 % of allowable deflection. Using ETABS program, a verification of the proposed procedure was carried out for load case 2 to check deflection and stress since this load case is controlled by stress. Figure (4.21) shows 84

Chapter (4)


the load case and figure (4.22) shows the maximum absolute deflection value which is 1 mm while MATLAB produced 0.99949 mm for nodes 17, 18, 19 and 20 in X and Y directions which is approximately matched. Figure (4.23) shows axial load in Load Case 2 where axial force in members 55, 56, 57 and 58 is 4.206 kip and its stress equals 167.34 MPa (24.27 ksi) where it was 24.28 from MATLAB which is approximately matched. As shown in table (4.13), the truss in load case (1) is controlled by deflection constrain while it is controlled by stress constrain in load case (2). So it is not possible to optimize topology for that truss considering these two load cases. Table (4.14) lists the results encountered in literature for the 72 member space truss problem compared with the results of the proposed approach. As illustrated from the results, the current proposed approach resulted in the most optimized value of fitness (weight) which is less than all results found in literature. Current study resulted size optimized truss weighs 375.77 lb, which represents 99.06% of the best result recorded in the literature Adeli [53].

85

Chapter (4)


Figure 4.21: Load case 2 for 72-bar truss structure in ETABS program (N).

86

Chapter (4)


Figure 4.22: Maximum deflection of load case 2 for 72-bar truss structure in ETABS program (mm).

87

Chapter (4)


Figure 4.23: Axial load in members due to Load Case 2 for 72-bar truss structure in ETABS program (kip).

88

Chapter (4)


Table 4.13: Members stresses for 72-bar space truss.

Member No.

Cross section group

Cross section Area (cm2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

A1 A1 A1 A1 A2 A2 A2 A2 A2 A2 A2 A2 A3 A3 A3 A3 A4 A4 A5 A5 A5 A5 A6 A6 A6 A6 A6 A6 A6 A6 A7 A7 A7 A7 A8 A8

12.865 12.865 12.865 12.865 3.717 3.717 3.717 3.717 3.717 3.717 3.717 3.717 0.434 0.434 0.434 0.434 0.143 0.143 10.268 10.268 10.268 10.268 3.572 3.572 3.572 3.572 3.572 3.572 3.572 3.572 0.193 0.193 0.193 0.193 0.232 0.232

Length (m)

Weight (kg)

152.40 152.40 152.40 152.40 340.77 340.77 340.77 340.77 340.77 340.77 340.77 340.77 304.80 304.80 304.80 304.80 431.06 431.06 152.40 152.40 152.40 152.40 340.77 340.77 340.77 340.77 340.77 340.77 340.77 340.77 304.80 304.80 304.80 304.80 431.06 431.06

5.42 5.42 5.42 5.42 3.51 3.51 3.51 3.51 3.51 3.51 3.51 3.51 0.37 0.37 0.37 0.37 0.17 0.17 4.33 4.33 4.33 4.33 3.37 3.37 3.37 3.37 3.37 3.37 3.37 3.37 0.16 0.16 0.16 0.16 0.28 0.28

89

Load Case 1 (Abs. value) Axial Stresses Forces (MPa) (KN)

23.09 7.47 31.09 7.47 7.96 5.96 4.85 6.05 6.05 4.85 7.96 5.96 0.18 0.18 0.18 0.18 0.13 0.04 12.54 5.47 22.24 5.47 5.83 8.27 5.74 5.03 5.03 5.74 5.83 8.27 0.27 0.09 0.09 0.27 0.22 0.40

17.94 5.81 24.17 5.81 21.42 16.04 13.04 16.28 16.28 13.04 21.42 16.04 4.10 4.10 4.10 4.10 9.31 3.10 12.22 5.33 21.66 5.33 16.32 23.17 16.07 14.07 14.07 16.07 16.32 23.17 13.83 4.61 4.61 13.83 9.59 17.27


22.55 22.55 22.55 22.55 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.49 0.49 0.49 0.49 0.18 0.18 21.31 21.31 21.31 21.31 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 0.13 0.13 0.13 0.13 0.18 0.18

17.53 17.53 17.53 17.53 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 11.26 11.26 11.26 11.26 12.41 12.41 20.75 20.75 20.75 20.75 2.86 2.86 2.86 2.86 2.86 2.86 2.86 2.86 6.91 6.91 6.91 6.91 7.68 7.68

Chapter (4)


Table 4.13: Members stresses for 72-bar space truss (continued).

Member No.

Cross section group

Cross section Area (cm2)

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

A9 A9 A9 A9 A10 A10 A10 A10 A10 A10 A10 A10 A11 A11 A11 A11 A12 A12 A13 A13 A13 A13 A14 A14 A14 A14 A14 A14 A14 A14 A15 A15 A15 A15 A16 A16

4.540 4.540 4.540 4.540 2.948 2.948 2.948 2.948 2.948 2.948 2.948 2.948 0.220 0.220 0.220 0.220 0.413 0.413 1.118 1.118 1.118 1.118 3.160 3.160 3.160 3.160 3.160 3.160 3.160 3.160 2.499 2.499 2.499 2.499 3.186 3.186

Total Weight (kg)

Length (m)

Weight (kg)

152.40 152.40 152.40 152.40 340.77 340.77 340.77 340.77 340.77 340.77 340.77 340.77 304.80 304.80 304.80 304.80 431.06 431.06 152.40 152.40 152.40 152.40 340.77 340.77 340.77 340.77 340.77 340.77 340.77 340.77 304.80 304.80 304.80 304.80 431.06 431.06

1.91 1.91 1.91 1.91 2.78 2.78 2.78 2.78 2.78 2.78 2.78 2.78 0.19 0.19 0.19 0.19 0.49 0.49 0.47 0.47 0.47 0.47 2.98 2.98 2.98 2.98 2.98 2.98 2.98 2.98 2.11 2.11 2.11 2.11 3.80 3.80 170.46

90



2.80 4.72 12.14 4.72 8.90 5.69 4.72 5.56 5.56 4.72 8.90 5.69 0.71 0.00 0.00 0.71 1.60 0.27 12.50 0.27 3.87 0.27 5.12 10.90 4.31 4.54 4.54 4.31 5.12 10.90 6.54 2.09 2.09 6.54 2.76 8.41 Max. Stress

23.44 23.44 23.44 23.44 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 0.98 0.98 0.98 0.98 1.87 1.87 18.73 18.73 18.73 18.73 3.91 3.91 3.91 3.91 3.91 3.91 3.91 3.91 1.87 1.87 1.87 1.87 2.36 2.36 Max. Stress

6.17 10.39 26.75 10.39 30.18 19.32 16.00 18.86 18.86 16.00 30.18 19.32 32.36 0.00 0.00 32.36 38.73 6.46 111.60 2.39 34.61 2.39 16.19 34.49 13.65 14.36 14.36 13.65 16.19 34.49 26.17 8.37 8.37 26.17 8.66 26.39 111.60

51.64 51.64 51.64 51.64 4.53 4.53 4.53 4.53 4.53 4.53 4.53 4.53 44.49 44.49 44.49 44.49 45.19 45.19 167.42 167.42 167.42 167.42 12.39 12.39 12.39 12.39 12.39 12.39 12.39 12.39 7.48 7.48 7.48 7.48 7.40 7.40 167.42

0.4791

0.1

0.1

0.1579

0.5501

0.3449

0.4984

379.31

A11

A12

A13

A14

A15

A16

Wight (Lbs.)

0.1

A7

A10

0.5689

A6

0.5137

1.1567

A5

A9

0.1

A4

0.1

0.1

A3

A8

0.5332

A2

91

380.32

0.6749

0.3981

0.5501

0.1557

0.1049

0.1

0.5227

0.5177

0.1

0.1

0.5028

1.283

0.1

0.1

0.4933

1.8562

1996

1986

2.0259

Cao [48]

Adeli [53]

A1

Design variable area (in2)

381.2

0.506

0.377

0.557

0.161

0.1

0.1

0.532

0.611

0.1

0.1

0.524

1.246

0.1

0.1

0.524

1.818

1971

Venkayya [54]

388.63

0.608

0.341

0.594

0.158

0.151

0.1

0.548

0.264

0.1

0.1

0.579

1.107

0.1

0.1

0.503

2.078

1974

Schmit [55]

387.67

0.5317

0.4378

0.5614

0.1519

0.1583

0.1

0.5273

0.5814

0.1

0.1

0.5244

1.2526

0.1

0.1

0.5259

1.8589

1979

383.12

0.103

0.122

0.525

1.91

0.103

0.11

0.498

1.31

0.111

0.103

0.535

0.535

0.521

0.379

0.544

0.161

2000

Khan et al Erbatur [56] et al. [26]

395.97

0.342

0.453

0.773

0.149

0.1

0.1

0.608

0.552

0.1

0.1

0.542

1.024

0.1

0.1

0.521

1.464

1971

Gellatly [57]

Table 4.14: Optimization results for 72-bar space truss.

384.1341

0.1

0.1

0.462033

1.862385

0.100075

0.102094

0.483267

1.315822

0.102617

0.1

0.573289

0.654788

0.726192

0.293503

0.577252

0.15346

2003

Barbaso [58]

406.03

0.508

0.35

0.587

0.174

0.174

0.174

0.477

0.508

0.174

0.174

0.587

0.978

0.174

0.174

0.587

2.046

2000

407.37

0.431

0.431

0.587

0.174

0.22

0.22

0.508

0.431

0.287

0.174

0.508

1.457

0.174

0.174

0.477

2.046

2003

Ayvaz Dede et al. [59] Disc. [60] Disc.

380.783

0.1

0.1

0.513

1.86

0.105

0.101

0.523

1.25

0.1

0.1

0.53

0.62

0.51

0.37

0.555

0.156

2009

379.85

0.5922

0.3922

0.5507

0.1565

0.1005

0.1004

0.5172

0.5209

0.1012

0.1

0.5269

1.2476

0.1

0.1

0.5059

1.8577

2007

381.91

0.1

0.1

0.5185

1.7427

0.1

0.1

0.5193

1.3079

0.1095

0.1

0.5464

0.5142

0.5619

0.4967

0.5092

0.1615

2007

Talaslioglu Camp [8] Perez [51] BB-BC [59] PSO BGAwEIS

382.35

0.604

0.442

0.604

0.154

0.118

0.109

0.55

0.505

0.1

0.1

0.469

1.288

0.1

0.1

0.496

1.702

2011

Tayfun [44]

375.77

0.49385

0.38727

0.48978

0.1733

0.064083

0.034095

0.45689

0.70363

0.03593

0.029919

0.5536

1.5916

0.022226

0.06734

0.57611

1.9941

2015

Current study

Chapter (4) Case Study And Results

Chapter (5)

Summary And Conclusions

Chapter (5) SUMMARY AND CONCLUSIONS 5.1 Summary When dealing with simultaneous size, shape and topology optimization of trusses, the number of possible solutions reaches great levels, which need long chromosome and long calculation time. In this thesis, an approach is proposed for optimization of plane and space trusses using GA method where GA design variables are changed to contain node deflections instead of member sections and connectivity in addition to nodal coordinates. The proposed procedure was applied to three of the classical truss benchmark problems to carry out sizing, topology and shape optimization and the results are investigated and compared with that found in previous works.

5.2 Conclusions Based on the conducted study, the following conclusions can be drawn:

92

Chapter (5)

1-


GA is considered a suitable tool for truss optimization with different categories and configurations.

2-

The proposed approach for selecting the design variables succeeded to reduce the chromosome length needed to represent truss variables that lead to reduction in number of possible solutions and the calculation time and effort by using a few numbers of chromosomes.

3-

As the truss cross sections are not included in the design variables, it is possible in the proposed approach to choose sections from huge numbers of a predetermined set without any increase in the chromosome length. The results were compared to the results of previous work found in literature where the presented results produce better optimum weight than previous results.

4-

As compared with literature, more optimized solutions are obtained

by

using

the

proposed

approach

with

less

computational costs. The number of bites used and number of possible solutions required to solve the problem is significantly reduced.

93

Chapter (5)

5-


The results obtained utilizing the proposed approach are more optimized when compared with previous research. Less weight was obtained for all studied benchmarks for combined optimization or even sizing optimization.

6-

The proposed GA technique allows full optimization (size, shape and topology) in the same time without increase in the consumed time.

7-

The proposed approach proved that GA still has the flexibility as a tool for structural optimization.

5.3 Suggestions for Future Work Based on the work in the present thesis and the results obtained, the following may be valid points for future research: 1.

To use more revolutionary techniques as the real coded GA, Particle Swarm optimization (PSO), Ant Colony optimization (ACO) and Big Bang - Big Crunch optimization (BB-BC).

2.

To apply these methods to more advanced structural types such as tensigrity structures.

94

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100

‫ﺟﺎﻣﻌـــــــــﺔ اﻟﺰﻗﺎزﯾﻖ‬ ‫ﻛﻠﯿــــﺔ اﻟﮭﻨﺪﺳــــــــــﺔ‬ ‫ﻗﺴﻢ اﻟﮭﻨﺪﺳﺔ اﻹ ﻧﺸﺎﺋﯿﺔ‬

‫أﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت اﻟﻔﺮاﻏﯿﺔ ﺑﺎﺳﺘﺨﺪام اﻟﺨﻮارزﻣﺎت اﻟﻮراﺛﯿﺔ‬ ‫رﺳﺎﻟﺔ ﻣﻘﺪﻣﺔ ﻣﻦ‬

‫اﻟﻤﮭﻨﺪس ‪ /‬ﻋﺜﻤﺎن ﺣﻤﺪى ﻋﺜﻤﺎن‬ ‫ﻟﻨﯿﻞ درﺟﺔ اﻟﻤﺎﺟﺴﺘﯿﺮ ﻓﻰ اﻟﻌﻠﻮم اﻟﮭﻨﺪﺳﯿﮫ ﻓﻲ اﻟﮭﻨﺪﺳﺔ اﻻﻧﺸﺎﺋﯿﺔ‬ ‫ﺗﺤﺖ إﺷــﺮاف‬

‫أ‪.‬د ‪/‬ﻋﺜﻤﺎن ﺷﻌﻼن‬ ‫أﺳﺘﺎذ اﻹﻧﺸﺎءات‪ -‬و رﺋﯿﺲ ﻗﺴﻢ اﻟﮭﻨﺪﺳﺔ اﻹﻧﺸﺎﺋﯿﺔ‬ ‫ﻛﻠﯿﺔ اﻟﮭﻨﺪﺳﺔ ‪ -‬ﺟﺎﻣﻌﺔ اﻟﺰﻗﺎزﯾﻖ‬

‫أ‪.‬د‪.‬م ‪ /‬ﺛﺮوت ﺻﻘﺮ‬

‫أ‪ .‬د ‪ /‬ﻋﺎطـــﻒ ﻋﺮاﻗﻲ ﺑﻜﺮى‬

‫أﺳﺘﺎذ ﻣﺴﺎﻋﺪ ﺑﻘﺴﻢ اﻟﮭﻨﺪﺳﺔ اﻹﻧﺸﺎﺋﯿﺔ‬ ‫ﻛﻠﯿﺔ اﻟﮭﻨﺪﺳـــــﺔ ‪ -‬ﺟﺎﻣﻌﺔ اﻟﺰﻗﺎزﯾﻖ‬

‫أﺳﺘﺎذ اﻹﻧﺸﺎءات‪ -‬ﻗﺴﻢ اﻟﮭﻨﺪﺳﺔ اﻹﻧﺸﺎﺋﯿﺔ‬ ‫ﻛﻠﯿﺔ اﻟﮭﻨﺪﺳـــــــﺔ ‪ -‬ﺟﺎﻣﻌﺔ اﻟﺰﻗﺎزﯾﻖ‬

‫‪٢٠١٥‬‬

‫ﻣﻠﺨﺺ اﻟﺮﺳﺎﻟﺔ‬ ‫ﯾﻤﻜﻦ ﺗﻄﻮﯾﺮ ﻋﻤﻠﯿﺔ أﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت وﺗﻼﻓﻰ ﻋﯿﻮب اﻟﻄﺮق اﻟﺘﻘﻠﯿﺪﯾﺔ ﻷﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت ﺑﺎﻻﺿﺎﻓﺔ‬ ‫اﻟﻰ ﺗﻘﻠﯿﻞ اﻟﻌﻤﻠﯿﺎت اﻟﺤﺴﺎﺑﯿﺔ اﻟﻤﺴﺘﺨﺪﻣﺔ وﺑﺎﻟﺘﺎﻟﻰ اﻟﻮﻗﺖ اﻟﻤﺴﺘﮭﻠﻚ ﻟﮭﺬه اﻟﺤﺴﺎﺑﺎت ﻣﻊ اﻟﺤﺼﻮل اﯾﻀﺎ‬ ‫ﻋﻠﻰ ﻧﺘﺎﺋﺞ اﻓﻀﻞ ﻣﻦ ﻧﺘﺎﺋﺞ اﻻﺑﺤﺎث اﻟﺴﺎﺑﻘﺔ ﻋﻦ طﺮﯾﻖ اﺳﺘﺨﺪام اﻧﻮاع ﺟﺪﯾﺪه ﻣﻦ اﻟﻤﺘﻐﯿﺮات ﻏﯿﺮ اﻟﺘﻰ‬ ‫ﺗﺴﺘﺨﺪم ﺑﺎﻟﻄﺮق اﻟﺘﻘﻠﯿﺪﯾﺔ ﺣﯿﺚ ﺗﻢ اﺳﺘﺒﺪال اﻟﻤﺘﻐﯿﺮات اﻟﺨﺎﺻﺔ ﺑﻘﻄﺎﻋﺎت اﻟﻌﻨﺎﺻﺮ اﻟﻤﺨﺘﻠﻔﺔ وﺣﺎﻻت‬ ‫ﺗﻮزﯾﻊ اﻟﻌﻨﺎﺻﺮ ﺑﯿﻦ اﻟﻨﻘﺎط اﻟﻤﺨﺘﻠﻔﺔ ﺑﻤﺘﻐﯿﺮ ﺧﺎص ﺑﺎزاﺣﺔ اﻟﻨﻘﺎط اﻟﺮﺋﯿﺴﯿﺔ دون ﻧﻘﺎط اﻟﺪﻋﺎﺋﻢ ﺑﺎﻻﺿﺎﻓﺔ‬ ‫اﻟﻰ اﺳﺘﺨﺪام ﻣﺘﻐﯿﺮ اﺣﺪاﺛﯿﺎت اﻟﻨﻘﺎط ﻣﺘﻐﯿﺮه اﻻﺣﺪاﺛﯿﺎت ﻟﻌﻤﻞ أﻣﺜﻠﯿﺔ ﻛﺎﻣﻠﺔ ﻟﻠﺠﻤﺎﻟﻮن ﻣﻦ ﺣﯿﺚ أﻣﺜﻠﯿﺔ‬ ‫ﻟﻠﻘﻄﺎﻋﺎت واﺣﺪاﺛﯿﺎت اﻟﻨﻘﺎط وﺗﻮزﯾﻊ اﻟﻌﻨﺎﺻﺮ‪.‬‬ ‫ﺗﺘﻜﻮن اﻟﺮﺳﺎﻟﺔ ﻣﻦ ﺧﻤﺴﮫ ﻓﺼﻮل ﺑﺎﻹﺿﺎﻓﺔ اﻟﻰ ﻣﻠﺨﺺ ﺑﺎﻟﻠﻐﺔ اﻟﻌﺮﺑﯿﺔ و ﻣﻠﺨﺺ ﺑﺎﻟﻠﻐﺔ اﻻﻧﺠﻠﯿﺰﯾﺔ‬ ‫وﻗﺎﺋﻤﺔ ﺑﺎﻟﻤﺮاﺟﻊ و ﻓﯿﻤﺎ ﯾﻠﻰ ﻧﺒﺬة ﻣﺨﺘﺼﺮة ﻋﻦ ﻛﻞ ﻓﺼﻞ ﻣﻦ ﻓﺼﻮل اﻟﺮﺳﺎﻟﺔ ‪:‬‬ ‫اﻟﻔﺼﻞ اﻻول‪:‬‬ ‫ﯾﺤﺘﻮى اﻟﻔﺼﻞ اﻷول ﻋﻠﻰ ﻣﻘﺪﻣﺔ ﻋﻦ أﻣﺜﻠﯿﺔ اﻟﻤﻨﺸﺂت ﺑﺼﻔﺔ ﻋﺎﻣﺔ ﻣﻮﺿﺤﺎ اھﻤﯿﺔ ھﺬه اﻷﻣﺜﻠﯿﺔ ﻓﻰ ﺗﻘﻠﯿﻞ‬ ‫وزن اﻟﻤﻨﺸﺄ وﺗﻜﻠﻔﺘﮫ واﻟﻤﺪة اﻟﺰﻣﻨﯿﺔ اﻟﻤﺴﺘﮭﻠﻜﺔ ﻓﻰ اﻧﺸﺎءه ﺛﻢ ﯾﺘﻄﺮق اﻟﻰ أﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت ﺛﻢ ﯾﻮﺿﺢ‬ ‫اﻟﺪاﻓﻊ واﻟﮭﺪف ﻣﻦ اﻟﺪراﺳﺔ ﺛﻢ ﻣﻠﺨﺺ ﻟﻠﺮﺳﺎﻟﮫ ‪.‬‬ ‫اﻟﻔﺼﻞ اﻟﺜﺎﻧﻰ‪:‬‬ ‫ﯾﺤﺘﻮى اﻟﻔﺼﻞ اﻟﺜﺎﻧﻲ ﻋﻠﻰ ﺗﻌﺮﯾﻒ ووﺻﻒ ﻟﻠﺠﻤﺎﻟﻮن وﻋﺮض ﻟﻼﻧﻮاع اﻟﻤﺨﺘﻠﻔﺔ ﻷﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت ﺛﻢ‬ ‫ﺷﺮح ﻟﻄﺮق اﻷﻣﺜﻠﯿﺔ اﻟﻤﺨﺘﻠﻔﺔ وﺧﺎﺻﺔ طﺮﯾﻘﺔ اﻟﺨﻮارزﻣﺎت اﻟﺠﯿﻨﯿﮫ ﻣﻊ ﺷﺮح ﻟﺨﻄﻮاﺗﮭﺎ واﻟﻤﺨﻄﻂ‬ ‫اﻟﻌﺎم ﻟﮭﺎ وﺷﺮح ﻟﻠﺘﻄﺒﯿﻘﺎت اﻟﮭﻨﺪﺳﯿﺔ اﻟﺘﻰ اﺳﺘﺨﺪﻣﺖ اﻷﻣﺜﻠﯿﺔ ﺑﺎﺳﺘﺨﺪام اﻟﺨﻮارزﻣﺎت اﻟﺠﯿﻨﯿﮫ ﻟﻠﻤﻨﺸﺂت‬ ‫ﺳﻮاء ﻟﻠﺠﻤﺎﻟﻮﻧﺎت او اﻻطﺎرات ﺑﺎﻻﺿﺎﻓﺔ اﻟﻰ اﻟﻤﻨﺸﺄت اﻟﻤﺮﻛﺒﺔ ‪.‬‬ ‫اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ‪:‬‬ ‫ﯾﺤﺘﻮى اﻟﻔﺼﻞ اﻟﺜﺎﻟﺚ ﻋﻠﻰ اﻟﻤﺘﻐﯿﺮات اﻟﻤﺴﺘﺨﺪﻣﺔ ﺑﺎﻟﻄﺮﯾﻘﺔ اﻟﺘﻘﻠﯿﺪﯾﺔ ﻷﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت وﻛﺬﻟﻚ‬ ‫اﻟﻌﻮاﻣﻞ اﻟﺘﻰ ﺗﻮﺛﺮ ﻋﻠﻰ طﻮل اﻟﻜﺮﻣﻮﺳﻮم اﻟﻤﺴﺘﺨﺪم ﻟﻌﻤﻠﯿﺔ اﻷﻣﺜﻠﯿﺔ ﺑﺎﺳﺘﺨﺪام اﻟﺨﻮارزﻣﺎت اﻟﺠﯿﻨﯿﺔ ﻣﻊ‬ ‫اﻟﺘﻄﺒﯿﻖ ﻋﻠﻰ ﻣﺜﺎل ﻟﺘﻮﺿﯿﺢ ﺣﺴﺎب طﻮل اﻟﻜﺮﻣﻮﺳﻮم اﻟﻼزم ﻟﻌﻤﻞ أﻣﺜﻠﯿﺔ ﻛﺎﻣﻠﺔ ﻟﮭﺬا اﻟﻤﺜﺎل ﺑﺎﻟﻄﺮﯾﻘﺔ‬ ‫اﻟﺘﻘﻠﯿﺪﯾﺔ ﻣﻊ ﺷﺮح ﻟﻠﻌﯿﻮب اﻟﺨﺎﺻﺔ ﺑﺄﺳﺘﺨﺪام ﺗﻠﻚ اﻟﻄﺮﯾﻘﺔ اﻟﺘﻘﻠﯿﺪﯾﺔ ﺛﻢ ﺷﺮح اﻟﻄﺮﯾﻘﺔ اﻟﻤﻘﺘﺮﺣﺔ ﻣﻊ‬ ‫ﺗﻮﺿﯿﺢ ﻟﻠﻤﺘﻐﯿﺮات اﻟﻤﺴﺘﺨﺪﻣﺔ ﺑﮭﺎ ﻣﻊ اﻟﺘﻄﺒﯿﻖ ﻋﻠﻰ ﻧﻔﺲ اﻟﻤﺜﺎل ﻟﺤﺴﺎب طﻮل اﻟﻜﺮﻣﻮﺳﻮم اﻟﻤﻄﻠﻮب‬ ‫ﻟﻌﻤﻞ أﻣﺜﻠﯿﺔ ﻛﺎﻣﻠﺔ ﻣﻊ ﻣﻘﺎرﻧﺘﮫ ﺑﻄﻮل اﻟﻜﺮﻣﻮﺳﻮم اﻟﺨﺎص ﺑﺎﻟﻄﺮﯾﻘﺔ اﻟﺘﻘﻠﯿﺪﯾﺔ ﻣﻊ ﺷﺮح ﺧﻄﻮات اﻟﻄﺮﯾﻘﺔ‬ ‫)‪( ١‬‬

‫اﻟﻤﻘﺘﺮﺣﺔ ﯾﻌﻘﺒﮭﺎ ﻣﻘﺎرﻧﺔ ﺑﯿﻦ اﻟﻄﺮﯾﻘﺘﯿﻦ ﻣﻮﺿﺤﺎ ﻣﻤﯿﺰات اﻟﻄﺮﯾﻘﺔ اﻟﻤﻘﺘﺮﺣﺔ ﻋﻠﻰ اﻟﻄﺮﯾﻘﺔ اﻟﺘﻘﻠﯿﺪﯾﺔ‬ ‫وﯾﻨﺘﮭﻰ اﻟﻔﺼﻞ ﺑﻮﺻﻒ ﻟﻠﻘﯿﻮد اﻟﻤﺴﺘﺨﺪﻣﺔ واﻟﻤﺨﻄﻂ اﻟﻌﺎم اﻟﻤﺴﺘﺨﺪم ﻟﻠﻄﺮﯾﻘﺔ اﻟﻤﻘﺘﺮﺣﺔ ‪.‬‬ ‫اﻟﻔﺼﻞ اﻟﺮاﺑﻊ‪:‬‬ ‫ﯾﺤﺘﻮى اﻟﻔﺼﻞ اﻟﺮاﺑﻊ ﻋﻠﻰ ﺗﻄﺒﯿﻖ ﻟﻠﻄﺮﯾﻘﺔ اﻟﻤﻘﺘﺮﺣﺔ ﻋﻠﻰ اﻟﻨﻤﺎذج اﻟﺸﮭﯿﺮة اﻟﺘﻰ ﺗﻢ اﺳﺘﺨﺪاﻣﮭﺎ ﻓﻰ‬ ‫اﻻﺑﺤﺎث اﻟﺴﺎﺑﻘﺔ ﻷﻣﺜﻠﯿﺔ اﻟﺠﻤﺎﻟﻮﻧﺎت وھﻰ ﺟﻤﺎﻟﻮن ﺛﻨﺎﺋﻰ اﻻﺑﻌﺎد ذو ‪ ١٠‬ﻋﻨﺎﺻﺮ وﺟﻤﺎﻟﻮن ﺛﻼﺛﻰ‬ ‫اﻻﺑﻌﺎد ذو ‪ ٢٥‬ﻋﻨﺼﺮ ﺑﺎﻻﺿﺎﻓﺔ اﻟﻰ ﺟﻤﺎﻟﻮن ﺛﻼﺛﻰ اﻻﺑﻌﺎد ذو ‪ ٧٢‬ﻋﻨﺼﺮ وﺗﻢ ﻣﻘﺎرﻧﺔ ﻧﺘﺎﺋﺞ اﻟﻄﺮﯾﻘﺔ‬ ‫اﻟﻤﻘﺘﺮﺣﺔ ﻣﻊ ﻧﺘﺎﺋﺞ اﻻﺑﺤﺎث اﻟﺴﺎﺑﻘﺔ ﺣﯿﺚ اﻋﻄﺖ اﻟﻄﺮﯾﻘﺔ اﻟﻤﻘﺘﺮﺣﺔ ﻧﺘﺎﺋﺞ اﻓﻀﻞ ﻣﻦ ﻧﺘﺎﺋﺞ اﻻﺑﺤﺎث‬ ‫اﻟﺴﺎﺑﻘﺔ ‪.‬‬ ‫اﻟﻔﺼﻞ اﻟﺨﺎﻣﺲ‪:‬‬ ‫ھﺬا اﻟﻔﺼﻞ ﯾﺘﻨﺎول ﻋﺮض ﻷھﻢ ﻧﺘﺎﺋﺞ اﻟﺮﺳﺎﻟﺔ و ﻛﺬﻟﻚ اﻟﺘﻮﺻﯿﺎت ﻷﺑﺤﺎث اﻟﻤﺴﺘﻘﺒﻠﯿﺔ‪.‬‬

‫)‪( ٢‬‬